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An improved test for interfacial shear strength
An improved test for interfacial shear strength J. C. SWEARENGEN and 7. F. COVERT
An improved test for determining the shear strength of the interfacial bond in fibre composite systems is presented. The procedure retains the direct interpretation of the fibre pullout technique, but by employing multiple fibres the tediousness of the latter method is avoided. Stainless steel fibres with several different matrices, both ductile and brittle are used, and the experimental observations are discussed with reference to related studies in the literature. It is shown that matrix ductility has a significant effect upon effective pullout stress and that pullout stress is insensitive to fibre spacing.
The concept of reinforcement of a weak matrix by strong fibres depends wholly upon the ability of the fibre-matrix interface to transfer loads from the matrix to the fibre. The strength of the interface determines composite properties when applied loads are biaxial or not aligned with the fibre direction and controls the behaviour of the system when flaws or cracks are present. Quantitative evaluation of interface strength is therefore a prerequisite to complete characterization of any composite system. In this paper a new technique is presented for determining interface strength and the strength is measured for several fibre-matrix systems. The influence of matrix properties on the experimental results is evaluated. Most of the techniques that have evolved to measure the strength of the interface are discussed in a review paper by Salkind.1 These techniques include 'direct' methods such as pullout tests and bonded-rod strength tests, or indirect methods such as measurement of the length of pulledout fibres at a fracture surface and study of the mechanical behaviour of discontinuous fibre composites as a function of fibre length. Interface strength can also be estimated from the fracture surface in a composite with continuous fibres if the fibres are brittle and well-characterized with respect to flaw distributions. 2,3 Each traditional technique falls short of the ideal. The pullout test requires preparation of many specimens and presents problems in gripping individual fibres for tension testing; the bonded-rod tests may not produce stress fields analogous to that experienced by fibres within a composite and the indirect methods require not only testing of completed composite samples and scanning-electron-microscopy of fracture surfaces, but also quantitative evaluation of the statistics of fibre flaw distribution. However, the pullout test does simulate conditions which occur during fracture Sandia Laboratories, Albuquerque, New Mexico 87115, USA
COMPOSITES
. SEPTEMBER
1973
of real composites. Effort has therefore been devoted to the development of an improved technique whereby the tediousness of the pullout test can be reduced without sacrificing any of its simulation features. The technique involves simply the combination of many pullout tests into one, ie, pullout of a bundle of fibres embedded to various lengths in the matrix. This configuration permits determination of the critical embedded length of fibre in one test instead of many, and also allows evaluation of the effect of interfibre spacing and matrix stress state on the fibre pullout stress. PROCEDURE
Thirteen bright-finished type 302 stainless steel wires were employed in each test. The filament diameter was 0.508 mm (0.020 in) with each fibre pre-straightened to facilitate handling. Average tensile strength of the fibres was determined from tensile tests to be about 1.86 x 109 N/m 2 (270 000 psi) and their fracture strain was less than 2%. The behaviour of the fibres in this condition was approximated by treating them as brittle. The grip in which pullout took place, termed the 'active' grip, was made with a 'V' facing the gauge section as shown in Fig. 1a. This symmetric arrangement produced a range of embedded fibre lengths and eliminated induced bending during loading. The fibre centre spacing was normally two fibre diameters, except that a distance of three diameters was used to study fibre spacing effects. Some of the grips were cast from liquids in metal moulds. These moulds were made with two halves to permit their removal from the specimen after curing or solidification of the grip material. The mating faces of the moulds were grooved to hold the fibres in place during solidification; the fibres were free to move axially in order to eliminate the possibility of fibre preloading. 'Resinbond Adhesive 907' epoxy and Pb-5Sb alloy grips were formed in this manner.
203
rip
Activegrip
4
_l
Doublertabs~.~ m
Doubler tabs
Fig.1 Specimen configuration (a) perspective view without loading tabs (b) side view showing testing arrangement
The fibres were degreased in trichlorethylene vapour before use, and for the lead system, the fibres were also electroplated with a one-micron coating of nickel and then precoated with lead before assembly. The mould faces were coated with a release agent to facilitate specimen removal, and in the case of the lead matrix the mould was preheated to eliminate any possibility of chill casting. The 'dead' grip had to be made longer than the active grip to ensure that pullout occurred in the latter. When a new system is first evaluated the critical length for pullout is unknown and must be estimated. Nevertheless, if the active grip is too long and all the fibres break, a lower bound to the interfacial strength is still obtained. Similarly, an upper
204
bound is produced if all the fibres pull out. Active grip length can easily be modified after specimen fabrication. Doubler tabs which extended beyond the grips by about 50 mm (2 in) were epoxy-bonded to the grips for tensile testing, Fig. lb. The tabs were line-drilled and mounted in the testing machine using clevis pins. All tensile testing was conducted at a crosshead rate of approximately 0.004 mm/s (0.01 in/min) while a continuous record of load versus displacement was produced. In addition to the epoxy and lead systems employed, a matrix material that permitted study of the effect of matrix ductility on pullout stress was also treated. This unique material consists of porous aluminium produced by the plasma-spray technique. 4 The sprayed material is produced with a density of about 84% of the theoretical value, and is capable of undergoing increases in density by a process of sintering at high vacuum 1.33 x 10 -4 N/m 2 (10 -6 mm Hg) and elevated temperature 643°C (1 190°F). This procedure reduces the yield strength and greatly enhances the ductility so that the matrix changes from a very brittle (fracture strain less than 1%) to a rather ductile state. 5 Pullout specimens as well as unreinforced sections of this material were subjected to a series of sintering treatments for different times. Pullout specimens were produced from the sprayed material by mounting the fibres in a frame resembling a loom and masking the gauge section with pieces of transite before building up the grips. Since this technique was not dimensionally controlled, flat faces were machined on the grips by grinding before the transite was removed. The gripping and loading procedure was identical to that employed for the epoxy and lead systems. For insight into pullout behaviour the properties of the matrix materials must also be known. Test specimens of plasma-sprayed aluminium were made for this purpose, in the form of thin-walled cylinders. The mechanical behaviour of the sprayed aluminium cylinders was studied by bonding the cylinders into gripping heads and loading in pure torsion in an MTS servohydraulic testing machine. A continuous record of torque versus shear strain was produced using foil strain gauges to measure the strain. The shear strength of the lead alloy matrix was determined by casting test specimens in a heated mould and tensionloading these in an Instron. The shear yield strength was taken as one-half of the tensile yield strength. The shear strength of the 907 epoxy resin was determined from a simple lap-shear test. ANAL
YSIS
For elastic deformation, the stress in each fibre is given by E~ (1)
oi Li
where E is Young's modulus of the fibres, b is the composite deflection, and L i is the free length of the ith fibre. The stress in any fibre is inversely proportional to its free length, so that the shortest (outer) fibres will be first to yield or fracture. The stresses could be equalized among the fibres i f L i were made equal and the fibres were embedded to different lengths in a rectangular grip. In that case, however, the load transferred to the matrix and to other fibres at the fibre ends would introduce an additional degree of redundancy into the problem.
COMPOSITES. SEPTEMBER 1973
6 (in)
The total tensile load borne by the composite is equal to the sum of individual fibre loads, and elongation is the same in each fibre. For elastic deformation the external load is therefore
P=ZPi
=
AE8 E ( L i )
"1
0-005
0"010
I
300-
(2)
200
0"020
I
1
E=200 x 103 MN/m 2 (29x106 psi) L = 7 0 + (2-6289)i mm (2.75+(0.1035)iin / l
where
i
0-015
I
. . . .
7 /
600
/
L/
800
"13 C
(0'
400
E
where A is the cross-sectional area of the fibre. Equations 1 and/or 2 permit determination of elastic fibre stresses as functions of either deflection or external load. After N elastic-plastic fibres have reached their yield stress the external load is given by
P'=Aoy[LiE(Lj)'I +N] /
In this equation the sum is to be taken over the unyielded fibres and L i becomes the free length of the longest yielded fibre. Similarly, after brittle fracture of N fibres the external load is P'=
P -
NAafi
(4)
lOG
Brief consideration reveals that the fibre stress of interest, with respect to interface strength evaluation, is bounded by the fibre yield or fracture strength, and the stress calculated for the adjacent unbroken, but pulled-out wire. The 'critical' length L c is bounded by the embedded lengths of the pulled-out fibre and the adjacent wire that failed. To a first approximation then, Equations 1 - 4 are unnecessary because the interracial shear stress can be determined from the usual force balance relationship
/
I
0.1
E/gstic -plastic fibres
1
0.2
I
-I I
I
0.3 0.4 8 (ram)
!00
I
0.5
Fig.2 Theoretically predicted load-elongation relation for multiple-fibre pullout specimen having elastic-plastic or brittle fibres
tions 1 and 5 show that the ratio of adjacent fibre free lengths must be Li+l < Li
Equations 1 - 4 were used to reproduce the composite loaddeflection curve and to determine the stress in each fibre at pullout. Fibre strength can be determined separately or from the composite limit load. For this study the fibre strength was determined by separate tensile tests and then used to predict composite load versus deflection for comparison to measured values. The foregoing equations were solved by computer; typical results are displayed in Fig.2 for elastic-plastic and brittle fibres having a strength of 1.1 x 109 N/m 2 (160 000 psi). The curve is inexact to the extent that fibre stress-strain behaviour is approximated, and fibre deflection within the grips is not included in L i.
o n
0 (3)
-g
D~" --
(6)
4rL c
EXPERIMENTAL
R E S U L TS
A pullout test recorded for stainless steel fibres in plasma sprayed aluminium is shown in Fig.3. In addition to the fibre stresses at failure, the friction stress for pullout can also be estimated from this trace by taking into account the change in embedded fibre length during the pullout process. Similar behaviour occurred in the other systems. Pullout in the epoxy-matrix system was accompanied by serrations, typical of interracial friction, in the ,loadelongation curve.
where D is fibre diameter and o represents either the yield or the fracture strength of a fibre. This relationship is based on the assumption of a uniform shear stress distribution and it will only yield a value for the strength of the interface directly if the stress is actually distributed in that way.
The fibres which pulled out of the 907 epoxy grip were essentially free of adhering resin. The weakest interfacial link is therefore the fibre-resin bond and not the epoxy matrix. The shear stress for bond failure determined by Equation 5 was 7.58-8.96 x 106 N/m 2 (1 100-1 300 psi) whereas the lap shear test for this brittle resin produced a failure stress of 24.132 x 106 N/m 2 (3 500 psi). The fibrematrix interface strength is similar in magnitude to values listed by Broutman 6 for metal-fibre reinforced epoxies, but much smaller than the value of 46.3 x 106 N/m 2 (6 720 psi) reported by Greszczuk 7 for aluminium rods in epoxy resin. The shear stress to pull fibres out of the lead alloy matrix was 8.96 x 10 6 N/m 2 (1 300 psi) which differs from the shear yield strength of the alloy by less than 689 x 103 N/m 2 (100 psi). After pullout, a film of matrix material remained adherent to the pulled out fibres, so that the pertinent shear strength given by Equation 5 should be close to that of the lead matrix, as observed.
If the fibres are spaced too far apart such that the difference in embedded lengths of adjacent fibres is large, the stress in the inner fibres may reach the magnitude for pullout before the outer fibres break. To avoid that situation, which produces an indeterminate fibre stress state, Equa-
Failure in the plasma-sprayed aluminium system occurred in the matrix as shown by the photograph in Fig.4. For this system, as well as the lead system, interface strength should be correlated with the properties of the matrix. In addition to the change in matrix properties during annealing,
oD r -
(5)
4L c
COMPOSITES . S E P T E M B E R 1973
205
the stainless steel Wires also softened so that their strength dropped from 1 862 to 896 x 106 N/m 2 (270 to 130 ksi) after 16 hours at 643°C (1 190°F) and their ductility rose. After two hours the elastic-plastic approximation better describes wire behaviour and the former is used in all calculations beyond two hours of annealing. The pullout stress determined from Equation 5 is shown in Fig.5 along with the yield and fracture strengths of the matrix. In the as-sprayed condition where the matrix is nearly brittle, the interface strength is considerably less than the strength of the matrix. After the sintering treatment to increase matrix ductility the interface strength becomes quite close to the shear yield strength of the matrix. Several tests were conducted in the stainless steel fibre-plasma sprayed aluminium system with a fibre centre distance of three, instead of two, fibre diameters. The results are included, with the rest of the data for aluminium-matrix tests at zero hours, in Fig.5. The pullout shear stresses for these tests clearly fell within the range of scatter of tests having the usual spacing.
Fibre pullout test rcsults~& AIr mvoutrix
t" ma~;,'x
b
"
_ matrix Cylinder • torsion results
, \Fracture
~ rU~
strength
por2usaluminum
of
s 4
20
.E
U 0 1:
U
trength of
I
porous aluminum 0
I
I
I
¢.-
I
Time (h) at 6 4 3 ° C (llgO°F) Fig.5 Interface shear strength for various composite systems determined by assuming a uniform shear stress distribution
DISCUSSION Deflection (in) 0"01 0.02 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6
300 250 /
~~Fibre
600
breaks
48O
200
A m
"10
o o .J
720
15C
300
I00
240
50
120 0"25
0"50 0"/5
I.O
1.25
"0 0 0 _1
1.50
Deflection (ram)
Fig.3 Experimentally observed load-elongation trace for multiplefibre pullout test in the stainless steel-aluminium system
The results obtained from these pullout tests are in accordance with the different behaviour expected in ductile and brittle matrix composites. Analytical treatment of the pullout problem for the case of an elastic matrix shows that the interfacial shear stress varies greatly along the length of the fibre. 7-9 Failure is expected to begin in the region of greatest stress, ie, at the points where the fibres enter the matrix and then to propagate along the interface. Behaviour of this sort may occur whether failure results from interface debonding, as in the epoxymatrix system, or occurs within the matrix, as in the assprayed aluminium system. The effect of non-uniform shear stress distribution is that the strength determined from Equation 5 will be smaller than the stress which initiates the failure. It should be noted that the stress distribution may also depend to some degree on the embedded length of the fibre. The shear stress concentration will be reduced when the matrix is ductile because plastic flow can relieve the stresses. The assumption of uniform shear stresses along the interface becomes a better approximation in that case and Equation 5 will yield values that agree more closely with the matrix properties. This behaviour is experimentally verified in the lead and sintered-aluminium matrix systems (Fig.5). The traditional one-dimensional elastic analysis of the pullout problem 7,9-10 neglects tensile stresses acting in a direction normal to the fibre axis. Such tensile stresses will be generated by lateral contraction of the fibres, however, and these tensile stresses could conceivably affect the interfacial failure mode. This seems especially probable for the case of propagation of a debond, although it is uncertain at this time whether propagation occurs by Type I, II, or III fracture mode. Existing analyses implicitly assume that debond occurs by propagation of a Type II (shear) crack. 7,9 The authors are aware of just one experiment treating the effect of normal stresses on pullout.11 It is most likely that this scarcity of data exists because three-dimensional stresses are not developed in the one-dimensional 'shear lag' model.
Fig.4 Photomicrograph of fibres pulled out of plasma-sprayed aluminium matrix, showing failure within the matrix
206
Interface strength values determined from the onedimensional analysis depend upon the quantity called
COMPOSITES
. SEPTEMBER
1973
interface 'width', which is usually taken as one-half the interfibre spacing. 7,10 The lack of agreement for interface strength in this study with that of Gre~zczuk 7 can be attributed in part to this factor. The interfacial shear stress in the shear lag model approaches a uniform distribution when fibre spacing becomes large, but the results of this study do not reveal a dependence of shear stress on spacing. This observation is supported by the results of the twodimensional analysis of the problem of a single fibre embedded in an elastic half-space. 8 In that case, in spite of an 'infinite' fibre spacing, the shear stress distribution has a similar functional form to the shear lag result for close fibre spacing. This observation holds everywhere along the interface except at the point where the fibre enters the matrix; at this point the shear stress must be zero to satisfy equilibrium requirements. The shear lag model apparently overemphasizes the influence of the spacing parameter because the foregoing observations indicate that the shear stress distribution is insensitive to fibre spacing.
3. The interface strength determined from the multiplefibre pullout test is independent of fibre spacing for the range of spacings investigated. This result is not in agreement with assumptions of the shear lag model.
A CKNO WL ED G EMEN TS
The authors would like to express their appreciation to W. R. Hoover and T. R. Guess of Sandia Laboratories for their critical review of the manuscript and also thank Dr A. Kelly for calling their attention to an additional pertinent reference. The work was supported and made possible by the United States Atomic Energy Commission.
REFERENCES
CONCL USIONS
An improved multiple-fibre pullout test for fibre-matrix interfacial shear strength has been developed and evaluated. Test results show that the method provides rapid determination o f these shear strengths, especially for the common case o f brittle fibres embedded in a ductile matrix. Several specific conclusions have resulted from the study: 1. The non-uniform interfacial shear stress distribution leads to fibre pullout from brittle matrices at fibre loads less than those that would occur under conditions of uniform shear stress. This result holds whether failure occurs in the fibre-matrix interface or within the matrix. 2. When failure occurs within the matrix, the interfacial shear stress concentration will decrease as the matrix ductility rises. In this case the interfacial shear strength approaches the yield strength of the matrix.
COMPOSITES. SEPTEMBER 1973
9 10 11
Salkind, M.J. 'The role of interfaces in fiber composites' Surfaces and Interfaces II: Physical and Mechanical Properties, Syracuse University Press (1968) 417 445 Tetelman, A. S. 'l.'racture processes in fiber composite materials' ASTM STP 460 (1969) 473 502 Cooper, G. A. 'The fracture toughness of composites reinforced with weakened fibers', J Mat Sci 5 (1970) 645 -654 Swearengen, J. C., Guess, T. R. 'Mechanical behaviour of metal-metal composite cylinders', 'Failure modes in composites', The Metallurgical Society" (1973) 149- 164 AIIred, R. E. Sandia l.aboratories, unpublished Broutman, L. J. 'Measurement of fiber-polymer matrix interface strength', ASTM STP452 (1968) 27 41 Greszczuk, L. B. 'Theoretical studies of the mechanics of the fiber-matrix interface in composites', ASTM STP 452 (1968) 42 58 Muki, R., Sternberg, 1"_. 'Elastostatic load-transfer to a half space from a partially embedded axially loaded rod' In1.1 Solids Structures 6 (1970) 69 90 Lawrence, P. 'Some theoretical considerations of fibre pullout from an elastic ma trix', J Mat Sci 7 (1972) 1-6 Cox, tt. L. 'The elasticity and strength of paper and other fibrous materials', Brit .I A ppl Phys 3 ( 1952) 72 79 Bowden, P. B. 'The effect of hydrostatic pressure on the fibre-matrix bond in a steel-resin model composite'..l Mat S c i 5 (1970) 517 520
207
An improved test for determining the shear strength of the interfacial bond in fibre composite systems is presented. The procedure retains the direct interpretation of the fibre pullout technique, but by employing multiple fibres the tediousness of the latter method is avoided. Stainless steel fibres with several different matrices, both ductile and brittle are used, and the experimental observations are discussed with reference to related studies in the literature. It is shown that matrix ductility has a significant effect upon effective pullout stress and that pullout stress is insensitive to fibre spacing.
The concept of reinforcement of a weak matrix by strong fibres depends wholly upon the ability of the fibre-matrix interface to transfer loads from the matrix to the fibre. The strength of the interface determines composite properties when applied loads are biaxial or not aligned with the fibre direction and controls the behaviour of the system when flaws or cracks are present. Quantitative evaluation of interface strength is therefore a prerequisite to complete characterization of any composite system. In this paper a new technique is presented for determining interface strength and the strength is measured for several fibre-matrix systems. The influence of matrix properties on the experimental results is evaluated. Most of the techniques that have evolved to measure the strength of the interface are discussed in a review paper by Salkind.1 These techniques include 'direct' methods such as pullout tests and bonded-rod strength tests, or indirect methods such as measurement of the length of pulledout fibres at a fracture surface and study of the mechanical behaviour of discontinuous fibre composites as a function of fibre length. Interface strength can also be estimated from the fracture surface in a composite with continuous fibres if the fibres are brittle and well-characterized with respect to flaw distributions. 2,3 Each traditional technique falls short of the ideal. The pullout test requires preparation of many specimens and presents problems in gripping individual fibres for tension testing; the bonded-rod tests may not produce stress fields analogous to that experienced by fibres within a composite and the indirect methods require not only testing of completed composite samples and scanning-electron-microscopy of fracture surfaces, but also quantitative evaluation of the statistics of fibre flaw distribution. However, the pullout test does simulate conditions which occur during fracture Sandia Laboratories, Albuquerque, New Mexico 87115, USA
COMPOSITES
. SEPTEMBER
1973
of real composites. Effort has therefore been devoted to the development of an improved technique whereby the tediousness of the pullout test can be reduced without sacrificing any of its simulation features. The technique involves simply the combination of many pullout tests into one, ie, pullout of a bundle of fibres embedded to various lengths in the matrix. This configuration permits determination of the critical embedded length of fibre in one test instead of many, and also allows evaluation of the effect of interfibre spacing and matrix stress state on the fibre pullout stress. PROCEDURE
Thirteen bright-finished type 302 stainless steel wires were employed in each test. The filament diameter was 0.508 mm (0.020 in) with each fibre pre-straightened to facilitate handling. Average tensile strength of the fibres was determined from tensile tests to be about 1.86 x 109 N/m 2 (270 000 psi) and their fracture strain was less than 2%. The behaviour of the fibres in this condition was approximated by treating them as brittle. The grip in which pullout took place, termed the 'active' grip, was made with a 'V' facing the gauge section as shown in Fig. 1a. This symmetric arrangement produced a range of embedded fibre lengths and eliminated induced bending during loading. The fibre centre spacing was normally two fibre diameters, except that a distance of three diameters was used to study fibre spacing effects. Some of the grips were cast from liquids in metal moulds. These moulds were made with two halves to permit their removal from the specimen after curing or solidification of the grip material. The mating faces of the moulds were grooved to hold the fibres in place during solidification; the fibres were free to move axially in order to eliminate the possibility of fibre preloading. 'Resinbond Adhesive 907' epoxy and Pb-5Sb alloy grips were formed in this manner.
203
rip
Activegrip
4
_l
Doublertabs~.~ m
Doubler tabs
Fig.1 Specimen configuration (a) perspective view without loading tabs (b) side view showing testing arrangement
The fibres were degreased in trichlorethylene vapour before use, and for the lead system, the fibres were also electroplated with a one-micron coating of nickel and then precoated with lead before assembly. The mould faces were coated with a release agent to facilitate specimen removal, and in the case of the lead matrix the mould was preheated to eliminate any possibility of chill casting. The 'dead' grip had to be made longer than the active grip to ensure that pullout occurred in the latter. When a new system is first evaluated the critical length for pullout is unknown and must be estimated. Nevertheless, if the active grip is too long and all the fibres break, a lower bound to the interfacial strength is still obtained. Similarly, an upper
204
bound is produced if all the fibres pull out. Active grip length can easily be modified after specimen fabrication. Doubler tabs which extended beyond the grips by about 50 mm (2 in) were epoxy-bonded to the grips for tensile testing, Fig. lb. The tabs were line-drilled and mounted in the testing machine using clevis pins. All tensile testing was conducted at a crosshead rate of approximately 0.004 mm/s (0.01 in/min) while a continuous record of load versus displacement was produced. In addition to the epoxy and lead systems employed, a matrix material that permitted study of the effect of matrix ductility on pullout stress was also treated. This unique material consists of porous aluminium produced by the plasma-spray technique. 4 The sprayed material is produced with a density of about 84% of the theoretical value, and is capable of undergoing increases in density by a process of sintering at high vacuum 1.33 x 10 -4 N/m 2 (10 -6 mm Hg) and elevated temperature 643°C (1 190°F). This procedure reduces the yield strength and greatly enhances the ductility so that the matrix changes from a very brittle (fracture strain less than 1%) to a rather ductile state. 5 Pullout specimens as well as unreinforced sections of this material were subjected to a series of sintering treatments for different times. Pullout specimens were produced from the sprayed material by mounting the fibres in a frame resembling a loom and masking the gauge section with pieces of transite before building up the grips. Since this technique was not dimensionally controlled, flat faces were machined on the grips by grinding before the transite was removed. The gripping and loading procedure was identical to that employed for the epoxy and lead systems. For insight into pullout behaviour the properties of the matrix materials must also be known. Test specimens of plasma-sprayed aluminium were made for this purpose, in the form of thin-walled cylinders. The mechanical behaviour of the sprayed aluminium cylinders was studied by bonding the cylinders into gripping heads and loading in pure torsion in an MTS servohydraulic testing machine. A continuous record of torque versus shear strain was produced using foil strain gauges to measure the strain. The shear strength of the lead alloy matrix was determined by casting test specimens in a heated mould and tensionloading these in an Instron. The shear yield strength was taken as one-half of the tensile yield strength. The shear strength of the 907 epoxy resin was determined from a simple lap-shear test. ANAL
YSIS
For elastic deformation, the stress in each fibre is given by E~ (1)
oi Li
where E is Young's modulus of the fibres, b is the composite deflection, and L i is the free length of the ith fibre. The stress in any fibre is inversely proportional to its free length, so that the shortest (outer) fibres will be first to yield or fracture. The stresses could be equalized among the fibres i f L i were made equal and the fibres were embedded to different lengths in a rectangular grip. In that case, however, the load transferred to the matrix and to other fibres at the fibre ends would introduce an additional degree of redundancy into the problem.
COMPOSITES. SEPTEMBER 1973
6 (in)
The total tensile load borne by the composite is equal to the sum of individual fibre loads, and elongation is the same in each fibre. For elastic deformation the external load is therefore
P=ZPi
=
AE8 E ( L i )
"1
0-005
0"010
I
300-
(2)
200
0"020
I
1
E=200 x 103 MN/m 2 (29x106 psi) L = 7 0 + (2-6289)i mm (2.75+(0.1035)iin / l
where
i
0-015
I
. . . .
7 /
600
/
L/
800
"13 C
(0'
400
E
where A is the cross-sectional area of the fibre. Equations 1 and/or 2 permit determination of elastic fibre stresses as functions of either deflection or external load. After N elastic-plastic fibres have reached their yield stress the external load is given by
P'=Aoy[LiE(Lj)'I +N] /
In this equation the sum is to be taken over the unyielded fibres and L i becomes the free length of the longest yielded fibre. Similarly, after brittle fracture of N fibres the external load is P'=
P -
NAafi
(4)
lOG
Brief consideration reveals that the fibre stress of interest, with respect to interface strength evaluation, is bounded by the fibre yield or fracture strength, and the stress calculated for the adjacent unbroken, but pulled-out wire. The 'critical' length L c is bounded by the embedded lengths of the pulled-out fibre and the adjacent wire that failed. To a first approximation then, Equations 1 - 4 are unnecessary because the interracial shear stress can be determined from the usual force balance relationship
/
I
0.1
E/gstic -plastic fibres
1
0.2
I
-I I
I
0.3 0.4 8 (ram)
!00
I
0.5
Fig.2 Theoretically predicted load-elongation relation for multiple-fibre pullout specimen having elastic-plastic or brittle fibres
tions 1 and 5 show that the ratio of adjacent fibre free lengths must be Li+l < Li
Equations 1 - 4 were used to reproduce the composite loaddeflection curve and to determine the stress in each fibre at pullout. Fibre strength can be determined separately or from the composite limit load. For this study the fibre strength was determined by separate tensile tests and then used to predict composite load versus deflection for comparison to measured values. The foregoing equations were solved by computer; typical results are displayed in Fig.2 for elastic-plastic and brittle fibres having a strength of 1.1 x 109 N/m 2 (160 000 psi). The curve is inexact to the extent that fibre stress-strain behaviour is approximated, and fibre deflection within the grips is not included in L i.
o n
0 (3)
-g
D~" --
(6)
4rL c
EXPERIMENTAL
R E S U L TS
A pullout test recorded for stainless steel fibres in plasma sprayed aluminium is shown in Fig.3. In addition to the fibre stresses at failure, the friction stress for pullout can also be estimated from this trace by taking into account the change in embedded fibre length during the pullout process. Similar behaviour occurred in the other systems. Pullout in the epoxy-matrix system was accompanied by serrations, typical of interracial friction, in the ,loadelongation curve.
where D is fibre diameter and o represents either the yield or the fracture strength of a fibre. This relationship is based on the assumption of a uniform shear stress distribution and it will only yield a value for the strength of the interface directly if the stress is actually distributed in that way.
The fibres which pulled out of the 907 epoxy grip were essentially free of adhering resin. The weakest interfacial link is therefore the fibre-resin bond and not the epoxy matrix. The shear stress for bond failure determined by Equation 5 was 7.58-8.96 x 106 N/m 2 (1 100-1 300 psi) whereas the lap shear test for this brittle resin produced a failure stress of 24.132 x 106 N/m 2 (3 500 psi). The fibrematrix interface strength is similar in magnitude to values listed by Broutman 6 for metal-fibre reinforced epoxies, but much smaller than the value of 46.3 x 106 N/m 2 (6 720 psi) reported by Greszczuk 7 for aluminium rods in epoxy resin. The shear stress to pull fibres out of the lead alloy matrix was 8.96 x 10 6 N/m 2 (1 300 psi) which differs from the shear yield strength of the alloy by less than 689 x 103 N/m 2 (100 psi). After pullout, a film of matrix material remained adherent to the pulled out fibres, so that the pertinent shear strength given by Equation 5 should be close to that of the lead matrix, as observed.
If the fibres are spaced too far apart such that the difference in embedded lengths of adjacent fibres is large, the stress in the inner fibres may reach the magnitude for pullout before the outer fibres break. To avoid that situation, which produces an indeterminate fibre stress state, Equa-
Failure in the plasma-sprayed aluminium system occurred in the matrix as shown by the photograph in Fig.4. For this system, as well as the lead system, interface strength should be correlated with the properties of the matrix. In addition to the change in matrix properties during annealing,
oD r -
(5)
4L c
COMPOSITES . S E P T E M B E R 1973
205
the stainless steel Wires also softened so that their strength dropped from 1 862 to 896 x 106 N/m 2 (270 to 130 ksi) after 16 hours at 643°C (1 190°F) and their ductility rose. After two hours the elastic-plastic approximation better describes wire behaviour and the former is used in all calculations beyond two hours of annealing. The pullout stress determined from Equation 5 is shown in Fig.5 along with the yield and fracture strengths of the matrix. In the as-sprayed condition where the matrix is nearly brittle, the interface strength is considerably less than the strength of the matrix. After the sintering treatment to increase matrix ductility the interface strength becomes quite close to the shear yield strength of the matrix. Several tests were conducted in the stainless steel fibre-plasma sprayed aluminium system with a fibre centre distance of three, instead of two, fibre diameters. The results are included, with the rest of the data for aluminium-matrix tests at zero hours, in Fig.5. The pullout shear stresses for these tests clearly fell within the range of scatter of tests having the usual spacing.
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Time (h) at 6 4 3 ° C (llgO°F) Fig.5 Interface shear strength for various composite systems determined by assuming a uniform shear stress distribution
DISCUSSION Deflection (in) 0"01 0.02 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6
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Fig.3 Experimentally observed load-elongation trace for multiplefibre pullout test in the stainless steel-aluminium system
The results obtained from these pullout tests are in accordance with the different behaviour expected in ductile and brittle matrix composites. Analytical treatment of the pullout problem for the case of an elastic matrix shows that the interfacial shear stress varies greatly along the length of the fibre. 7-9 Failure is expected to begin in the region of greatest stress, ie, at the points where the fibres enter the matrix and then to propagate along the interface. Behaviour of this sort may occur whether failure results from interface debonding, as in the epoxymatrix system, or occurs within the matrix, as in the assprayed aluminium system. The effect of non-uniform shear stress distribution is that the strength determined from Equation 5 will be smaller than the stress which initiates the failure. It should be noted that the stress distribution may also depend to some degree on the embedded length of the fibre. The shear stress concentration will be reduced when the matrix is ductile because plastic flow can relieve the stresses. The assumption of uniform shear stresses along the interface becomes a better approximation in that case and Equation 5 will yield values that agree more closely with the matrix properties. This behaviour is experimentally verified in the lead and sintered-aluminium matrix systems (Fig.5). The traditional one-dimensional elastic analysis of the pullout problem 7,9-10 neglects tensile stresses acting in a direction normal to the fibre axis. Such tensile stresses will be generated by lateral contraction of the fibres, however, and these tensile stresses could conceivably affect the interfacial failure mode. This seems especially probable for the case of propagation of a debond, although it is uncertain at this time whether propagation occurs by Type I, II, or III fracture mode. Existing analyses implicitly assume that debond occurs by propagation of a Type II (shear) crack. 7,9 The authors are aware of just one experiment treating the effect of normal stresses on pullout.11 It is most likely that this scarcity of data exists because three-dimensional stresses are not developed in the one-dimensional 'shear lag' model.
Fig.4 Photomicrograph of fibres pulled out of plasma-sprayed aluminium matrix, showing failure within the matrix
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Interface strength values determined from the onedimensional analysis depend upon the quantity called
COMPOSITES
. SEPTEMBER
1973
interface 'width', which is usually taken as one-half the interfibre spacing. 7,10 The lack of agreement for interface strength in this study with that of Gre~zczuk 7 can be attributed in part to this factor. The interfacial shear stress in the shear lag model approaches a uniform distribution when fibre spacing becomes large, but the results of this study do not reveal a dependence of shear stress on spacing. This observation is supported by the results of the twodimensional analysis of the problem of a single fibre embedded in an elastic half-space. 8 In that case, in spite of an 'infinite' fibre spacing, the shear stress distribution has a similar functional form to the shear lag result for close fibre spacing. This observation holds everywhere along the interface except at the point where the fibre enters the matrix; at this point the shear stress must be zero to satisfy equilibrium requirements. The shear lag model apparently overemphasizes the influence of the spacing parameter because the foregoing observations indicate that the shear stress distribution is insensitive to fibre spacing.
3. The interface strength determined from the multiplefibre pullout test is independent of fibre spacing for the range of spacings investigated. This result is not in agreement with assumptions of the shear lag model.
A CKNO WL ED G EMEN TS
The authors would like to express their appreciation to W. R. Hoover and T. R. Guess of Sandia Laboratories for their critical review of the manuscript and also thank Dr A. Kelly for calling their attention to an additional pertinent reference. The work was supported and made possible by the United States Atomic Energy Commission.
REFERENCES
CONCL USIONS
An improved multiple-fibre pullout test for fibre-matrix interfacial shear strength has been developed and evaluated. Test results show that the method provides rapid determination o f these shear strengths, especially for the common case o f brittle fibres embedded in a ductile matrix. Several specific conclusions have resulted from the study: 1. The non-uniform interfacial shear stress distribution leads to fibre pullout from brittle matrices at fibre loads less than those that would occur under conditions of uniform shear stress. This result holds whether failure occurs in the fibre-matrix interface or within the matrix. 2. When failure occurs within the matrix, the interfacial shear stress concentration will decrease as the matrix ductility rises. In this case the interfacial shear strength approaches the yield strength of the matrix.
COMPOSITES. SEPTEMBER 1973
9 10 11
Salkind, M.J. 'The role of interfaces in fiber composites' Surfaces and Interfaces II: Physical and Mechanical Properties, Syracuse University Press (1968) 417 445 Tetelman, A. S. 'l.'racture processes in fiber composite materials' ASTM STP 460 (1969) 473 502 Cooper, G. A. 'The fracture toughness of composites reinforced with weakened fibers', J Mat Sci 5 (1970) 645 -654 Swearengen, J. C., Guess, T. R. 'Mechanical behaviour of metal-metal composite cylinders', 'Failure modes in composites', The Metallurgical Society" (1973) 149- 164 AIIred, R. E. Sandia l.aboratories, unpublished Broutman, L. J. 'Measurement of fiber-polymer matrix interface strength', ASTM STP452 (1968) 27 41 Greszczuk, L. B. 'Theoretical studies of the mechanics of the fiber-matrix interface in composites', ASTM STP 452 (1968) 42 58 Muki, R., Sternberg, 1"_. 'Elastostatic load-transfer to a half space from a partially embedded axially loaded rod' In1.1 Solids Structures 6 (1970) 69 90 Lawrence, P. 'Some theoretical considerations of fibre pullout from an elastic ma trix', J Mat Sci 7 (1972) 1-6 Cox, tt. L. 'The elasticity and strength of paper and other fibrous materials', Brit .I A ppl Phys 3 ( 1952) 72 79 Bowden, P. B. 'The effect of hydrostatic pressure on the fibre-matrix bond in a steel-resin model composite'..l Mat S c i 5 (1970) 517 520
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