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Localized stress redistribution after fibre fracture in brittle matrix composites
PII: s1359-835x(%)oo133-9
ELSEVIER
Compositrs Part A 28A (1997) 347-353 0 1997 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-835x/97/$17.00
Localized stress redistribution after fibre fracture in brittle matrix composites
J. M. Martinez-Esnaola”, M. R. Elizalde, I. Puente
A. Martin-Meizoso, and M. Fuentes
A. M. Daniel,
J. M. SBnchez,
Centro de Estudios e lnvestigaciones Tknicas de Guipcizcoa (CEIT), Paseo de Manuel LardizBbal1520009 San Sebastiiin, Spain and Escuela Superior de lngenieros Industriales, Universidad de Plavarra, Apartado 1674, 20080 San Sebastih, Spain (Received 17 May 1996; revised 20 September 1996)
A model is detailed that describes the redistribution of stress, after individual fibre failure, amongst the intact reinforcing fibres of a ceramic matrix composite. The load drop carried by a single fibre when it breaks is balanced by extra loads in the intact fibres which are calculated using elastic solutions and variational mechanics. The magnitude of stress redistribution is shown to follow a d-” relationship, where d is the distance from the broken fibre and m depends on the fibre volume fraction and a non-dimensional parameter that normalizes the influence of the stress and the elastic properties of the composite. The model has been applied to two composites, Sic/Sic and CAS/SiC. It is shown that the stress redistribution is localized in both composites, the stress increments in the closest intact fibres being more than one order of magnitude greater than predicted by the assumption of global stress redistribution. 0 1997 Elsevier Science Limited. All rights reserved. (Keywords:
stress redistribution; damage localization; brittle matrix composites)
INTRODUCTION Continuous fibre-reinforced ceramic matrix composites (CMCs) have received much interest over the past 10 years as potential materials for high temperature structural applications in, for example, gas turbine engines for aircraft. The inherently brittle ceramic matrix is reinforced by long ceramic fibres that prevent brittle, catastrophic failure via the introduction of a number of toughening mechanisms. As long as the strain to failure of the fibres is greater than that of the matrix and the fibre-matrix interface bond is sufficiently weak, the mechanisms of matrix cracking, frictional sliding between fibre and matrix, and fibre pull-out’ all operate to pro-
duce a tough composite material. The work of fracture of such composites can be orders of magnitude greater than that of the monolithic matrix material. Under tensile stresses, CMCs are characterized by matrix cracking occurring at lower tensile strains than failure of the reinforcing fibres2. After matrix cracking, the load is carried by the reinforcing fibres that undergo frictional slip relative to the matrix and, when the stresses are high enough to cause individual fibre fractures, fibre pull-out from the matrix. Subsequent to matrix cracking, _~_._ ___*To whom correspondence should be addressed (at CEIT)
the material integrity is dependent on the nature of fibre failure at higher applied stresses. The work of fracture of the composite is greatly increased if the failure of fibres is steady or controlled, as this produces a large amount of evenly distributed fibre pull-out. Once an individual fibre fails, the load is redistributed among the other intact fibres. It is often assumed that this redistribution of load is global, i.e. the stress increment is equal for all the intact fibres3-j. However, it is reasonable to expect that the load is redistributed locally to the neighbouring fibres according to some function of their distance from the broken fibre6,7. Theoretically, a global load redistribution should result in stable tensile fracture of a unidirectionally reinforced composite yarn and a fracture morphology of evenly distributed pull-out length8’. Experimentally, however, unstable failure is observed, and fibre pull-out lengths are not randomly distributed, as concentrations of long pull-out lengths are often observed at the corners of tensile test-pieces. Therefore, the widely used global load redistribution assumption predicts behaviour that is not observed experimentally and therefore suggests that stress redistribution after fibre failure is in fact a localized phenomenon. This results in localized, uncontrolled failure and thus lower values of the total energy absorbed by the component.
347
Localized redistribution of fibre stress: J. M. Martinez-Esnaola
In this paper, the problem of the stress redistribution after failure of an individual fibre is solved using variational mechanics and superposition of well established solutions in elastostatics. A model is presented that describes the extent of localized stress redistribution in a CMC due to individual fibre failure in the region of a matrix crack.
et al.
of the load redistribution on the fibres closest to the broken fibre. For elastic analyses, the solution of the problem can be found by applying the principle of minimum complementary potential energy. For a statically admissible stress field (Tthe complementary potential energy II* is defined by839: I-I*(a) =
W*(o)dV -
V
ANALYSIS An approach is taken that considers the load drop on a broken fibre as load P, applied by a cylindrical indentor that has the radius of the broken fibre, acting on the surface of the composite material above and below the crack plane. Figure I shows a schematic of the model (only half the system is represented, due to symmetry), where L is a fictitious clamping length that accounts for the region where load transfer between fibre and matrix takes place, as described below. The load drop P is balanced by the extra loads carried by the intact fibres, which in turn produce additional reverse indentations on the matrix surface, with corresponding displacements wi. The distribution of such extra loads in the intact fibres constitutes the objective of the present analysis. The distribution of axial stresses in the fibres is simplified, as shown in Figure 2, through the definition of an equivalent clamping length L along which the axial stress in the fibre is assumed to be constant. The value of L is defined so that the area under the stress profile equals that corresponding to the actual stress distribution, therefore resulting in the same elongation of the fibre. If a constant frictional shear stress T is assumed at the interface, then this condition leads to a clamping length L equal to the transfer length (see Figure 2). Note that, although the friction between fibre and matrix is accounted for in the definition of L, the effect of the matrix layer along the clamping length is neglected in this approach. It may be anticipated that this will lead to a slight overestimate Crack plane
Liz
1
I
W
- - Matrix
Fibre i Figure 1 Load drop in the broken fibre modelled as an indentation on the composite. Smaller reverse indentations are produced by the extra loads carried by the intact fibres
348
J
J t-u
dS
S”
(1)
where o is the stress tensor, W* is the complementary strain energy density, V is the volume of the body, 5, is that part of the surface subjected to prescribed displacements U, and t = CPn is the reaction vector on S,, n being the outward unit normal. In this problem, S, is null. In order to evaluate the complementary potential energy II*, the fibres and the matrix will be considered separately. For a uniaxial stress state in a linear elastic material, as assumed in the fibres, W” reduces to c22/2Ef, where u is the axial stress and Ef is the Young’s modulus of the fibres. Therefore the complementary potential energy of the fibres II;-*can be written as:
(4 where N is the number of fibres, L is the clamping length in the simplified model (see Figure 2), and Ai and Pi are the cross-sectional area and the tensile load carried by the i-th fibre, respectively. For convenience in the formulation and without loss of generality, it will be assumed that the broken fibre is the N-th fibre. Then PN = -P is the load drop in the broken fibre which constitutes the external load applied to the body. Note that the spurious term corresponding to the broken fibre included in II; does not affect the application of the principle of minimum complementary potential energy, as this term acts as a constant for the minimizing process with respect to the unknowns Pi, i = 1,2,. . . , N - 1. The complementary potential energy of the matrix II;$ can be evaluated in an indirect manner using Clapeyron’s theorem for linear elastic materials. For the present problem, this states that the work done by the loads Pi acting through the corresponding displacements in the matrix wi is equal to twice the complementary strain energy. Therefore n&=2~~PiWj=~PiWi i=I
i=l
where the factor of 2 is included to account for the contributions of the composite material above and below the crack plane (see Figure 2). The calculation of the displacements Wi induced by the loads Pi in a heterogeneous anisotropic material of finite dimensions is rather complicated and no analytical solution exists. For the purpose of the present model, the displacements wi at the matrix surface can be approximated by superposition of the Boussinesq solution for a point load acting on the surface of a semi-infinite homogeneous
Localized
redistribution
of fibre stress: J. M. Martinez-Esnaolo
et al.
clamping length, L I
I matrix
I
4 1111111
1 1 1 1 1 1 1 lo Model for constant interfacial frictional shear stress Figure 2 Distribution are neglected
of axial stress in the fibres and definition
isotropic
as:
solid”‘“, wi =
0.54(1 - V2)Pi
i=1,2,...,N
ERi
neglected
Simplification of equivalent
clamping
length. The two matrix layers at both sides of the matrix crack
Finally, from (2) and (5) the complementary energy of the system becomes:
potential
N (1 - V2)Pi + c TEd.. ’ v j=l jfi (4)
where E is the Young’s modulus of the composite, u is its Poisson’s ratio, Ri is the radius of fibre i and dij is the distance between the centres of fibres i and j. The first term in the right-hand side of equation (4) represents the average displacement of the loaded area due to a uniformly distributed load in the fibre, and the second term is the contribution due to the loads applied by all the other fibres which are modelled as point loads. Introducing (4) into (3) the complementary potential energy of the matrix can be written as:
The equilibrium
condition N c 1x1
Pi = 0
(7)
is incorporated into the principle of minimum complementary potential energy using the Lagrange multipliers technique. Then, using (6) and (7) the function to be minimized is given by:
349
et al.
Localized redistribution of fibre stress: J. M. Martinez-Esnaola
rearranged in matrix form as: . .
Ml,N-I
1
M21
M22
. .
M2,N-1
1
MN-l,1 L
i= I,2 )“‘) N-
Ml2
MN-l,2
. .
1
..
1
MN-l,N-1
1
1
p2
1
pN-l
0
x*
l/‘&v 1/‘h
I
and then the introduction (9) yields
Pl
*
where X stands for the Lagrange multiplier. The condition of minimum for Q requires that $0,
Ml1
of equation (8) into equation
(16)
=P
l/k1,N LP. --‘ + ErAi
l-V2
1
E
where
i=1,2,...,N-
1
(10)
i= 1,2 ,..., N-1
M,=e+y)n,
The composite elastic modulus E is calculated as E =fEf
+ (1 -f)E,
(11)
wherefis the fibre volume fraction in the composite, and Ef and Em are the Young’s moduli of fibre and matrix, respectively. Assuming a constant frictional shear stress at the interface r, and neglecting the effect of residual stresses, the mean transfer length L is given by (see Figure
1 4=-q
(17) i,j = 1,2 , . . . , N-
l,i#j
The system of linear equations (16) can be solved for the N unknowns of the problem, i.e. Pi, i = 1,2, . . . , N - 1, and the Lagrange multiplier X’, using any standard numerical method.
2):
RESULTS AND DISCUSSION where R is the mean fibre radius, 0 is the remote stress applied to the composite andf, is the fibre volume fraction in the crack plane. Note that, in general, f, will be different fromf, as the number of intact fibres decreases during the process of fibre failure. Using (12) the system of equations (10) can be rewritten as:
i=
1,2,...,N-
1
(13)
where
E -fcEf 47 (1 - v2)f,Ef
p=Z
(14)
TEX x* = 2(1 - V2)
Note that if f, = f (as in the fracture of the first fibre), then using (11)
p=a
Cl-f)Em
47 (1 -
.v2)fEf
(15)
Equations (13) together with the equilibrium condition (7) provide a system of N simultaneous linear equations for the N unknowns of the problem, i.e. Pi, i = 1,2,. . . , N - 1, and the modified Lagrange multiplier X*. Note that, in this approach, P, = -P is the load drop in the broken fibre. Equations (13) and (7) can be
350
The first application of the model is the simulation of load redistribution in Sic/Sic yarns of two-dimensional woven composites produced by SEP (France). Typically, the yarn is composed of 500 fibres distributed in an ellipse12113.The distribution of SIC (Nicalon) fibre radii has been measured14, yielding a mean value of 7.12 pm with a standard deviation of 0.98pm. These data have been used to generate 500 fibres distributed at random inside an ellipse14 of semi-axes 613 and 91 pm, which results in a fibre volume fraction off = 0.46 15. The mean radius in the population of fibres R and the distances dij are calculated directly from this distribution. The broken fibre is then selected at random. The relevant material parameters for the analysis, which have been determined using nano-indentation techniques’s, are Ef = 209 GPa, Em = 487 GPa and r = 100 MPa; the Poisson’s ratio is taken13 as v = 0.2. As a representative stress level, a remote stress c = 210 MPa has been applied to the composite, as reported for the stress in the composite at the onset of matrix cracking12. The results are presented in terms of the normalized stresses in the fibres, which are defined as: q=a’=I ON
P. A N P Ai’
i=
1,2,...,N-
1
(18)
It is clear from equation (13) that a geometrical scaling of the problem does not affect the solution for the loads Pi and the corresponding normalized stresses. Therefore the results will be presented as a function of
Localized redistribution of fibre stress: J. M. Martinez- Esnaolo et al.
the distance from the broken fibre normalized by the mean radius of the fibres (d/R). Figure 3 shows the stress in the intact fibres as a function of the normalized distance from the broken fibre. Note that a global load redistribution would produce normalized stresses of about l/500 = 0.002. Therefore, it is clear that large underestimates of the fibre loads (of more than one order of magnitude) result from the global load redistribution assumption for the fibres in the vicinity of the broken fibre. In this example, the fibre closest to the broken fibre carries -8% of the load drop, i.e. about 40 times that corresponding to a global load redistribution. It is also apparent that the stress distribution in the intact fibres closest to the broken fibre can be described in the form c 0: d-“, where d is the distance from the broken fibre and m is given by the slope of the linear part of the curve in the log-log plot of Figure 3. For practical purposes, m has been determined by linear regression over the 10% of the fibres closest to the broken fibre. In this case, m = 2.4, in good agreement with the value of about 5/2 reported by the authors for this particular problem using a boundary collocation technique7 which involves an iterative procedure and lengthy calculations, instead of the variational mechanics analysis and the closed-form solution presented here for the stress redistribution. The model can be applied to simulate the stress redistribution for other geometries. As an example, Sic/Sic yarns with different fibre volume fractions have been modelled by generating distributions of variable numbers of fibres in the ellipse of semi-axes 613 pm x 91 pm. As a limit case of fibre packing, a fibre distribution with f’ - 0.9 has also been generated, which corresponds to a compact packing geometry where each fibre is surrounded by and in contact with another six fibres of the same radius. Figure 4 shows the resulting values of the exponent of stress redistribution m in the vicinity of the broken fibre as a function of the fibre volume fraction. The different symbols in Figure 4 correspond to the selection of different broken fibres. Although some influence of this selection can be noticed, this seems to represent a minor effect for the purposes of this work and will be ruled out for the rest of the calculations. During the process of successive ruptures of the fibres that lead to the final failure of the component, the Young’s modulus of the composite E can be regarded as a constant in the analysis, i.e. it is independent of the number of intact fibres in the matrix crack plane. Figure 5 compares the local redistribution exponents obtained with this assumption (using the value of Ecorresponding to an initial fibre volume fraction of 0.5) with those obtained using a Young’s modulus that varies with the fibre volume fraction, which represent the first fibre failure in a yarn with a given fibre volume fraction. The difference in values using both assumptions is not significant. The curve corresponding to a constant value of E represents an estimate of the evolution of the local redistribution exponent (from right to left) as failure of the fibres takes place. It can be seen that the load
0.1000
ii?
r
0.0100
%
Global load
P 4
1 2
0.0010
0.0001
10
1
__~~~__ _L
100
Normalized distance to the broken fibre (d/R) Figure 3 Variation broken fibre (W/Sic,
of stress in intact fibres with distance ,f = 0.4615, 0 = 2 10 MPa)
from the
-7---v-
3.5 r 3.0
I 2.5 g E
2.0
t g
1.5
ltI 1.0
0.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.6
0.9
1
D
Fibre volume fraction Figure 4 Exponent of stress redistribution volume fraction. Different symbols correspond ferent broken fibres
0.5
as a function of fibre to the selection of dif-
0 E =
constant
?E ?=fEf+(l-VEm J 0.2
0.3
0.4
6.5
Fibre volume fraction Figure 5 Comparison of local redistribution exponent for: (i) constant Young’s modulus (during process of successive fibre failure); and (ii) Young’s modulus varying with fibre volume fraction (first fibre failure)
redistribution is highly localized at the beginning (f; = 0.5) and tends to be more global as the number of intact fibres decreases. The limit situation of global load sharing is only attained for very low values of fibre volume fraction.
351
Localized redistribution of fibre stress: J. M. Martinez-Esnaola
The effect of the model parameters on the local redistribution exponent m is discussed below. To rationalize the influence of the fibre radii and the geometrical distribution of the fibres, consider the simple case where all the fibres have the same radius, Ri = R. Then equation (13) gives:
.i#i
i= 1,2,...,N-
1
(19)
where the terms dq/R are directly related to the fibre volume fractionf. As indicated above and in Figure 4, the influence of the position of the broken fibre can be neglected, and therefore the results can be presented as a function of the non-dimensional parameters ,8 and J: Figure 6 shows the local redistribution exponent as a function of ,/3 for different fibre volume fractions. The results indicate that global load redistribution is favoured by small values off(last stages of fibre failure) and high values of p (a stiff matrix and/or high (T/T ratio). The map of Figure 6 can be used to estimate the degree of localization of the load redistribution as a function of the non-dimensional parameters f and ,0; the parameter p encompasses the influence of the elastic constants Ef, E, and v, the interfacial shear stress 7, and the applied remote stress (T. As an example, the positions of two composite materials (two-dimensional woven Sic/Sic and O”/90”cross-ply CAS/SiC) are indicated on the map. The Sic/Sic properties are as above. The properties used for the 0” plies of CASjSiC are16-18 Ef = 209 GPa, E, = 97GPa, r = 16MPa, v = 0.2 and f = 0.39, and a remote stress of g = 6.5MPa was applied. In both cases, m N 2.4 for the first fibre fractures. The thick dashed line of Figure 6 represents the evolution of a Sic/Sic yarn during the process of successive ruptures of the fibres that lead to the final failure. Finally, note that the presence of other matrix cracks (multiple cracking of the matrix) further reduces the
3.5 ,
""I
et al.
Young’s modulus of the composite (stiffness of the composite) and therefore the value of /3. Then one might speculate that the present solution represents a lower bound for the localization of stresses in the fibres closest to the broken fibre. However, note that neglecting the influence of the matrix layer along the clamping length is expected to overestimate the stress localization, as discussed above. Therefore no definite conclusion can be stated in this respect at the present stage of the model.
CONCLUSIONS A model based on variational mechanics has been developed to assess how the stress drop in a broken fibre redistributes among the neighbouring intact fibres that are bridging a matrix crack. It has been determined that the extra loads in the fibres closest to the broken fibre behave as d-m, where d is the distance to the broken fibre. Values of m - 2.4 have been found for the first fibre fractures in Sic/Sic and CASjSiC composites, which indicate that the load redistribution is highly localized as opposed to the normal global load sharing assumption. The model predicts that the extra loads carried by the intact fibres in the vicinity of the broken fibre may be more than one order of magnitude higher than those corresponding to a global load redistribution assumption. This in turn may result in damage localization and consequently in decreased work of fracture of the composite. The influence of the material and geometrical constants can be rationalized in terms of two non-dimensional parameters: the fibre volume fraction and a parameter ,8 that is related to the elastic properties of the composite. This allows the construction of a map that can be used to estimate the local redistribution effect for a given material as a function of these two parameters. It is concluded that high values of p (stiff matrix and/or low values of the interfacial shear stress) and small fibre volume fractions tend to favour a global load redistribution and therefore high work of fracture.
I
ACKNOWLEDGEMENTS 3.0
2.5 ^s E
2.0
;
1.5
:: w
1.0
10
Beta Map showinglocal redistributionexponentas a functionof non-dimensionalparameter p and fibre volume fraction. The thick dashedlinesketchesthe evolutionof a SiC/SiCyam duringthe process of successiverupturesof the fibres
Figure 6
352
This work has been performed within the framework of the BRITE-EURAM project BE-5462 with financial support from the European Commission, co-ordinated by Rolls-Royce plc (UK), in collaboration with SNECMA (France) and SEP (France). Financial support of CEIT by Rolls-Royce plc, SNECMA and SEP is also gratefully acknowledged. The funding received from the Spanish ‘Comisi6n Interministerial de Ciencia y Tecnologia (CICYT)’ and from the ‘Viceconsejeria de Educacibn, Universidades e Investigacibn’ of the Basque Government made possible the purchase of the nano-indentation equipment. A.M.D., M.R.E. and I.P. are grateful to the Commission of the European Union, Directorate General XII for Science, Research and Development, to the Spanish Ministry of Education and Science, and to the
Localized
Department of Education, Universities and Research of the Basque Government, respectively, for the grants received.
redistribution
9. 10. Il. 12.
REFERENCES I. 2.
3.
4.
5. 6.
I.
8.
Evans, A.G. and Marshall, D.B., The mechanical behavior of ceramic matrix composites. Acta MetaN., 1989, 37, 2567-2583. Aveston, J., Cooper, G.A. and Kelly, A., The Properties of Fibre Composites, National Physical Laboratory. IPC Science and Technology Press, Guildford, UK, 1971, pp. 15-26. Budiansky, B., Hutchinson, J.W. and Evans, A.G., Matrix fracture in fibre-reinforced ceramics. J. Mech. Phys. Solids, 1986,34. 167. 189. Thouless, M.D. and Evans, A.G., Effect of pull-out on the mechanical properties of ceramic-matrix composites. ACIU Metall., 198836, 517-522. Curtin, W.A., Theory of mechanical properties of ceramicmatrix composites. J. Am. Ceram. Sot., 1991, 74, 2837-2845. Martin-Meizoso, A., Martinez-Esnaola, J.M., Sanchez, J.M., Puente, I., Elizalde, M.R., Daniel, A.M. and Fuentes, M., Modelling the tensile fracture behaviour of the reinforcing fibre yarns in brittle matrix composites. Fatigue Fract. Eng. Muter. Sfruct., submitted for publication. Martin-Meizoso, A., Martinez-Esnaola, J.M., Daniel, A.M., Sanchez, J.M., Puente, I. and Elizalde, M.R., Load transfer in ceramic matrix composites. In?. J. Fruct., 1996, 76, R55-R60. Gurtin, M.E., The linear theory of elasticity. In Mechanics of Solids, Vol. II, ed. C. Truesdell. Springer-Verlag, Berlin, 1984, pp. I m-295.
13.
14.
15.
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et al.
Kanninen, M.F. and Popelar, C.H.. Advanced Fracture Mechanics. Oxford University Press, New York, 1985. Theory and Applications. Pergamon Saada, AS., Elasticity. Press, Oxford, 1974. Westergaard, H.M., Theory of Elasticity and Plusticity. Dover Publications Inc., New York, 1952. Munier, E., Renard, J. and Boussuge, M., Finite element simulation of the mechanical behavior of monolayer bidirectional woven Sic/Sic composites. In High Temperature Ceram. Mutri.v Composites, Proc. HT-CMCI. eds. R. Naslain, J. Lamon and D. Doumeingts. Woodhead Publishing Ltd, Cambridge. 1993, pp. 5577564. Labrugere, C., Guette, A., Naslain, R.. Guillaumat, L. and Lamon, J., Etude des composites 2D Sic/Sic et SiC/MAS-L, Laboratoire des Composites Thermostructuraux, Pessac, 1994. Elizalde, M.R., Gil-Sevillano, J., Martin-Meizoso, A., Martinez-Esnaola, J.M. and Sanchez, J.M., BRITE-EURAM Project BE-5462. Reoort MAT1205.Tll. CEIT. 1995. Sanchez, J.M., Elizalde, M.R., Daniel, A.M., Martinez-Esnaola, J.M.. Puente, I. and Martin-Meizoso, A., Interfacial characterisation of 2D woven Sic/Sic and cross-ply 0“/90” CASjSiC composites. Composites Part A, 1996, 27A, 7817792. Beyerle, D.S., Spearing, S.M. and Evans, A.G., Damage mcchanisms and the mechanical properties of a laminated O/90 ceramic/ matrix composite. J. Am. Cerum. Sot., 1992, 75, 3321-3330. Elizalde, M.R., Sanchez, J.M., Daniel, A.M., Puente, I., MartinMeizoso. A., Martinez-Esnaola, J.M. and Fuentes, M., Evolucion del daiio en materiales compuestos de matriz ceramica. .&ales de Mecanicu de la Fructura, 1996, 13, 2499253. Daniel. A.M., Elizalde, M.R., Martinez-Esnaola, J.M. and Janczak, J., Residual stress and Poisson expansion effects on the fibre-matrix interfacial properties measured in fibre push tests. Cerum. Eng. Sci. Proc.. 1996, 17(4 5). 225 -232.
353
ELSEVIER
Compositrs Part A 28A (1997) 347-353 0 1997 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-835x/97/$17.00
Localized stress redistribution after fibre fracture in brittle matrix composites
J. M. Martinez-Esnaola”, M. R. Elizalde, I. Puente
A. Martin-Meizoso, and M. Fuentes
A. M. Daniel,
J. M. SBnchez,
Centro de Estudios e lnvestigaciones Tknicas de Guipcizcoa (CEIT), Paseo de Manuel LardizBbal1520009 San Sebastiiin, Spain and Escuela Superior de lngenieros Industriales, Universidad de Plavarra, Apartado 1674, 20080 San Sebastih, Spain (Received 17 May 1996; revised 20 September 1996)
A model is detailed that describes the redistribution of stress, after individual fibre failure, amongst the intact reinforcing fibres of a ceramic matrix composite. The load drop carried by a single fibre when it breaks is balanced by extra loads in the intact fibres which are calculated using elastic solutions and variational mechanics. The magnitude of stress redistribution is shown to follow a d-” relationship, where d is the distance from the broken fibre and m depends on the fibre volume fraction and a non-dimensional parameter that normalizes the influence of the stress and the elastic properties of the composite. The model has been applied to two composites, Sic/Sic and CAS/SiC. It is shown that the stress redistribution is localized in both composites, the stress increments in the closest intact fibres being more than one order of magnitude greater than predicted by the assumption of global stress redistribution. 0 1997 Elsevier Science Limited. All rights reserved. (Keywords:
stress redistribution; damage localization; brittle matrix composites)
INTRODUCTION Continuous fibre-reinforced ceramic matrix composites (CMCs) have received much interest over the past 10 years as potential materials for high temperature structural applications in, for example, gas turbine engines for aircraft. The inherently brittle ceramic matrix is reinforced by long ceramic fibres that prevent brittle, catastrophic failure via the introduction of a number of toughening mechanisms. As long as the strain to failure of the fibres is greater than that of the matrix and the fibre-matrix interface bond is sufficiently weak, the mechanisms of matrix cracking, frictional sliding between fibre and matrix, and fibre pull-out’ all operate to pro-
duce a tough composite material. The work of fracture of such composites can be orders of magnitude greater than that of the monolithic matrix material. Under tensile stresses, CMCs are characterized by matrix cracking occurring at lower tensile strains than failure of the reinforcing fibres2. After matrix cracking, the load is carried by the reinforcing fibres that undergo frictional slip relative to the matrix and, when the stresses are high enough to cause individual fibre fractures, fibre pull-out from the matrix. Subsequent to matrix cracking, _~_._ ___*To whom correspondence should be addressed (at CEIT)
the material integrity is dependent on the nature of fibre failure at higher applied stresses. The work of fracture of the composite is greatly increased if the failure of fibres is steady or controlled, as this produces a large amount of evenly distributed fibre pull-out. Once an individual fibre fails, the load is redistributed among the other intact fibres. It is often assumed that this redistribution of load is global, i.e. the stress increment is equal for all the intact fibres3-j. However, it is reasonable to expect that the load is redistributed locally to the neighbouring fibres according to some function of their distance from the broken fibre6,7. Theoretically, a global load redistribution should result in stable tensile fracture of a unidirectionally reinforced composite yarn and a fracture morphology of evenly distributed pull-out length8’. Experimentally, however, unstable failure is observed, and fibre pull-out lengths are not randomly distributed, as concentrations of long pull-out lengths are often observed at the corners of tensile test-pieces. Therefore, the widely used global load redistribution assumption predicts behaviour that is not observed experimentally and therefore suggests that stress redistribution after fibre failure is in fact a localized phenomenon. This results in localized, uncontrolled failure and thus lower values of the total energy absorbed by the component.
347
Localized redistribution of fibre stress: J. M. Martinez-Esnaola
In this paper, the problem of the stress redistribution after failure of an individual fibre is solved using variational mechanics and superposition of well established solutions in elastostatics. A model is presented that describes the extent of localized stress redistribution in a CMC due to individual fibre failure in the region of a matrix crack.
et al.
of the load redistribution on the fibres closest to the broken fibre. For elastic analyses, the solution of the problem can be found by applying the principle of minimum complementary potential energy. For a statically admissible stress field (Tthe complementary potential energy II* is defined by839: I-I*(a) =
W*(o)dV -
V
ANALYSIS An approach is taken that considers the load drop on a broken fibre as load P, applied by a cylindrical indentor that has the radius of the broken fibre, acting on the surface of the composite material above and below the crack plane. Figure I shows a schematic of the model (only half the system is represented, due to symmetry), where L is a fictitious clamping length that accounts for the region where load transfer between fibre and matrix takes place, as described below. The load drop P is balanced by the extra loads carried by the intact fibres, which in turn produce additional reverse indentations on the matrix surface, with corresponding displacements wi. The distribution of such extra loads in the intact fibres constitutes the objective of the present analysis. The distribution of axial stresses in the fibres is simplified, as shown in Figure 2, through the definition of an equivalent clamping length L along which the axial stress in the fibre is assumed to be constant. The value of L is defined so that the area under the stress profile equals that corresponding to the actual stress distribution, therefore resulting in the same elongation of the fibre. If a constant frictional shear stress T is assumed at the interface, then this condition leads to a clamping length L equal to the transfer length (see Figure 2). Note that, although the friction between fibre and matrix is accounted for in the definition of L, the effect of the matrix layer along the clamping length is neglected in this approach. It may be anticipated that this will lead to a slight overestimate Crack plane
Liz
1
I
W
- - Matrix
Fibre i Figure 1 Load drop in the broken fibre modelled as an indentation on the composite. Smaller reverse indentations are produced by the extra loads carried by the intact fibres
348
J
J t-u
dS
S”
(1)
where o is the stress tensor, W* is the complementary strain energy density, V is the volume of the body, 5, is that part of the surface subjected to prescribed displacements U, and t = CPn is the reaction vector on S,, n being the outward unit normal. In this problem, S, is null. In order to evaluate the complementary potential energy II*, the fibres and the matrix will be considered separately. For a uniaxial stress state in a linear elastic material, as assumed in the fibres, W” reduces to c22/2Ef, where u is the axial stress and Ef is the Young’s modulus of the fibres. Therefore the complementary potential energy of the fibres II;-*can be written as:
(4 where N is the number of fibres, L is the clamping length in the simplified model (see Figure 2), and Ai and Pi are the cross-sectional area and the tensile load carried by the i-th fibre, respectively. For convenience in the formulation and without loss of generality, it will be assumed that the broken fibre is the N-th fibre. Then PN = -P is the load drop in the broken fibre which constitutes the external load applied to the body. Note that the spurious term corresponding to the broken fibre included in II; does not affect the application of the principle of minimum complementary potential energy, as this term acts as a constant for the minimizing process with respect to the unknowns Pi, i = 1,2,. . . , N - 1. The complementary potential energy of the matrix II;$ can be evaluated in an indirect manner using Clapeyron’s theorem for linear elastic materials. For the present problem, this states that the work done by the loads Pi acting through the corresponding displacements in the matrix wi is equal to twice the complementary strain energy. Therefore n&=2~~PiWj=~PiWi i=I
i=l
where the factor of 2 is included to account for the contributions of the composite material above and below the crack plane (see Figure 2). The calculation of the displacements Wi induced by the loads Pi in a heterogeneous anisotropic material of finite dimensions is rather complicated and no analytical solution exists. For the purpose of the present model, the displacements wi at the matrix surface can be approximated by superposition of the Boussinesq solution for a point load acting on the surface of a semi-infinite homogeneous
Localized
redistribution
of fibre stress: J. M. Martinez-Esnaolo
et al.
clamping length, L I
I matrix
I
4 1111111
1 1 1 1 1 1 1 lo Model for constant interfacial frictional shear stress Figure 2 Distribution are neglected
of axial stress in the fibres and definition
isotropic
as:
solid”‘“, wi =
0.54(1 - V2)Pi
i=1,2,...,N
ERi
neglected
Simplification of equivalent
clamping
length. The two matrix layers at both sides of the matrix crack
Finally, from (2) and (5) the complementary energy of the system becomes:
potential
N (1 - V2)Pi + c TEd.. ’ v j=l jfi (4)
where E is the Young’s modulus of the composite, u is its Poisson’s ratio, Ri is the radius of fibre i and dij is the distance between the centres of fibres i and j. The first term in the right-hand side of equation (4) represents the average displacement of the loaded area due to a uniformly distributed load in the fibre, and the second term is the contribution due to the loads applied by all the other fibres which are modelled as point loads. Introducing (4) into (3) the complementary potential energy of the matrix can be written as:
The equilibrium
condition N c 1x1
Pi = 0
(7)
is incorporated into the principle of minimum complementary potential energy using the Lagrange multipliers technique. Then, using (6) and (7) the function to be minimized is given by:
349
et al.
Localized redistribution of fibre stress: J. M. Martinez-Esnaola
rearranged in matrix form as: . .
Ml,N-I
1
M21
M22
. .
M2,N-1
1
MN-l,1 L
i= I,2 )“‘) N-
Ml2
MN-l,2
. .
1
..
1
MN-l,N-1
1
1
p2
1
pN-l
0
x*
l/‘&v 1/‘h
I
and then the introduction (9) yields
Pl
*
where X stands for the Lagrange multiplier. The condition of minimum for Q requires that $0,
Ml1
of equation (8) into equation
(16)
=P
l/k1,N LP. --‘ + ErAi
l-V2
1
E
where
i=1,2,...,N-
1
(10)
i= 1,2 ,..., N-1
M,=e+y)n,
The composite elastic modulus E is calculated as E =fEf
+ (1 -f)E,
(11)
wherefis the fibre volume fraction in the composite, and Ef and Em are the Young’s moduli of fibre and matrix, respectively. Assuming a constant frictional shear stress at the interface r, and neglecting the effect of residual stresses, the mean transfer length L is given by (see Figure
1 4=-q
(17) i,j = 1,2 , . . . , N-
l,i#j
The system of linear equations (16) can be solved for the N unknowns of the problem, i.e. Pi, i = 1,2, . . . , N - 1, and the Lagrange multiplier X’, using any standard numerical method.
2):
RESULTS AND DISCUSSION where R is the mean fibre radius, 0 is the remote stress applied to the composite andf, is the fibre volume fraction in the crack plane. Note that, in general, f, will be different fromf, as the number of intact fibres decreases during the process of fibre failure. Using (12) the system of equations (10) can be rewritten as:
i=
1,2,...,N-
1
(13)
where
E -fcEf 47 (1 - v2)f,Ef
p=Z
(14)
TEX x* = 2(1 - V2)
Note that if f, = f (as in the fracture of the first fibre), then using (11)
p=a
Cl-f)Em
47 (1 -
.v2)fEf
(15)
Equations (13) together with the equilibrium condition (7) provide a system of N simultaneous linear equations for the N unknowns of the problem, i.e. Pi, i = 1,2,. . . , N - 1, and the modified Lagrange multiplier X*. Note that, in this approach, P, = -P is the load drop in the broken fibre. Equations (13) and (7) can be
350
The first application of the model is the simulation of load redistribution in Sic/Sic yarns of two-dimensional woven composites produced by SEP (France). Typically, the yarn is composed of 500 fibres distributed in an ellipse12113.The distribution of SIC (Nicalon) fibre radii has been measured14, yielding a mean value of 7.12 pm with a standard deviation of 0.98pm. These data have been used to generate 500 fibres distributed at random inside an ellipse14 of semi-axes 613 and 91 pm, which results in a fibre volume fraction off = 0.46 15. The mean radius in the population of fibres R and the distances dij are calculated directly from this distribution. The broken fibre is then selected at random. The relevant material parameters for the analysis, which have been determined using nano-indentation techniques’s, are Ef = 209 GPa, Em = 487 GPa and r = 100 MPa; the Poisson’s ratio is taken13 as v = 0.2. As a representative stress level, a remote stress c = 210 MPa has been applied to the composite, as reported for the stress in the composite at the onset of matrix cracking12. The results are presented in terms of the normalized stresses in the fibres, which are defined as: q=a’=I ON
P. A N P Ai’
i=
1,2,...,N-
1
(18)
It is clear from equation (13) that a geometrical scaling of the problem does not affect the solution for the loads Pi and the corresponding normalized stresses. Therefore the results will be presented as a function of
Localized redistribution of fibre stress: J. M. Martinez- Esnaolo et al.
the distance from the broken fibre normalized by the mean radius of the fibres (d/R). Figure 3 shows the stress in the intact fibres as a function of the normalized distance from the broken fibre. Note that a global load redistribution would produce normalized stresses of about l/500 = 0.002. Therefore, it is clear that large underestimates of the fibre loads (of more than one order of magnitude) result from the global load redistribution assumption for the fibres in the vicinity of the broken fibre. In this example, the fibre closest to the broken fibre carries -8% of the load drop, i.e. about 40 times that corresponding to a global load redistribution. It is also apparent that the stress distribution in the intact fibres closest to the broken fibre can be described in the form c 0: d-“, where d is the distance from the broken fibre and m is given by the slope of the linear part of the curve in the log-log plot of Figure 3. For practical purposes, m has been determined by linear regression over the 10% of the fibres closest to the broken fibre. In this case, m = 2.4, in good agreement with the value of about 5/2 reported by the authors for this particular problem using a boundary collocation technique7 which involves an iterative procedure and lengthy calculations, instead of the variational mechanics analysis and the closed-form solution presented here for the stress redistribution. The model can be applied to simulate the stress redistribution for other geometries. As an example, Sic/Sic yarns with different fibre volume fractions have been modelled by generating distributions of variable numbers of fibres in the ellipse of semi-axes 613 pm x 91 pm. As a limit case of fibre packing, a fibre distribution with f’ - 0.9 has also been generated, which corresponds to a compact packing geometry where each fibre is surrounded by and in contact with another six fibres of the same radius. Figure 4 shows the resulting values of the exponent of stress redistribution m in the vicinity of the broken fibre as a function of the fibre volume fraction. The different symbols in Figure 4 correspond to the selection of different broken fibres. Although some influence of this selection can be noticed, this seems to represent a minor effect for the purposes of this work and will be ruled out for the rest of the calculations. During the process of successive ruptures of the fibres that lead to the final failure of the component, the Young’s modulus of the composite E can be regarded as a constant in the analysis, i.e. it is independent of the number of intact fibres in the matrix crack plane. Figure 5 compares the local redistribution exponents obtained with this assumption (using the value of Ecorresponding to an initial fibre volume fraction of 0.5) with those obtained using a Young’s modulus that varies with the fibre volume fraction, which represent the first fibre failure in a yarn with a given fibre volume fraction. The difference in values using both assumptions is not significant. The curve corresponding to a constant value of E represents an estimate of the evolution of the local redistribution exponent (from right to left) as failure of the fibres takes place. It can be seen that the load
0.1000
ii?
r
0.0100
%
Global load
P 4
1 2
0.0010
0.0001
10
1
__~~~__ _L
100
Normalized distance to the broken fibre (d/R) Figure 3 Variation broken fibre (W/Sic,
of stress in intact fibres with distance ,f = 0.4615, 0 = 2 10 MPa)
from the
-7---v-
3.5 r 3.0
I 2.5 g E
2.0
t g
1.5
ltI 1.0
0.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.6
0.9
1
D
Fibre volume fraction Figure 4 Exponent of stress redistribution volume fraction. Different symbols correspond ferent broken fibres
0.5
as a function of fibre to the selection of dif-
0 E =
constant
?E ?=fEf+(l-VEm J 0.2
0.3
0.4
6.5
Fibre volume fraction Figure 5 Comparison of local redistribution exponent for: (i) constant Young’s modulus (during process of successive fibre failure); and (ii) Young’s modulus varying with fibre volume fraction (first fibre failure)
redistribution is highly localized at the beginning (f; = 0.5) and tends to be more global as the number of intact fibres decreases. The limit situation of global load sharing is only attained for very low values of fibre volume fraction.
351
Localized redistribution of fibre stress: J. M. Martinez-Esnaola
The effect of the model parameters on the local redistribution exponent m is discussed below. To rationalize the influence of the fibre radii and the geometrical distribution of the fibres, consider the simple case where all the fibres have the same radius, Ri = R. Then equation (13) gives:
.i#i
i= 1,2,...,N-
1
(19)
where the terms dq/R are directly related to the fibre volume fractionf. As indicated above and in Figure 4, the influence of the position of the broken fibre can be neglected, and therefore the results can be presented as a function of the non-dimensional parameters ,8 and J: Figure 6 shows the local redistribution exponent as a function of ,/3 for different fibre volume fractions. The results indicate that global load redistribution is favoured by small values off(last stages of fibre failure) and high values of p (a stiff matrix and/or high (T/T ratio). The map of Figure 6 can be used to estimate the degree of localization of the load redistribution as a function of the non-dimensional parameters f and ,0; the parameter p encompasses the influence of the elastic constants Ef, E, and v, the interfacial shear stress 7, and the applied remote stress (T. As an example, the positions of two composite materials (two-dimensional woven Sic/Sic and O”/90”cross-ply CAS/SiC) are indicated on the map. The Sic/Sic properties are as above. The properties used for the 0” plies of CASjSiC are16-18 Ef = 209 GPa, E, = 97GPa, r = 16MPa, v = 0.2 and f = 0.39, and a remote stress of g = 6.5MPa was applied. In both cases, m N 2.4 for the first fibre fractures. The thick dashed line of Figure 6 represents the evolution of a Sic/Sic yarn during the process of successive ruptures of the fibres that lead to the final failure. Finally, note that the presence of other matrix cracks (multiple cracking of the matrix) further reduces the
3.5 ,
""I
et al.
Young’s modulus of the composite (stiffness of the composite) and therefore the value of /3. Then one might speculate that the present solution represents a lower bound for the localization of stresses in the fibres closest to the broken fibre. However, note that neglecting the influence of the matrix layer along the clamping length is expected to overestimate the stress localization, as discussed above. Therefore no definite conclusion can be stated in this respect at the present stage of the model.
CONCLUSIONS A model based on variational mechanics has been developed to assess how the stress drop in a broken fibre redistributes among the neighbouring intact fibres that are bridging a matrix crack. It has been determined that the extra loads in the fibres closest to the broken fibre behave as d-m, where d is the distance to the broken fibre. Values of m - 2.4 have been found for the first fibre fractures in Sic/Sic and CASjSiC composites, which indicate that the load redistribution is highly localized as opposed to the normal global load sharing assumption. The model predicts that the extra loads carried by the intact fibres in the vicinity of the broken fibre may be more than one order of magnitude higher than those corresponding to a global load redistribution assumption. This in turn may result in damage localization and consequently in decreased work of fracture of the composite. The influence of the material and geometrical constants can be rationalized in terms of two non-dimensional parameters: the fibre volume fraction and a parameter ,8 that is related to the elastic properties of the composite. This allows the construction of a map that can be used to estimate the local redistribution effect for a given material as a function of these two parameters. It is concluded that high values of p (stiff matrix and/or low values of the interfacial shear stress) and small fibre volume fractions tend to favour a global load redistribution and therefore high work of fracture.
I
ACKNOWLEDGEMENTS 3.0
2.5 ^s E
2.0
;
1.5
:: w
1.0
10
Beta Map showinglocal redistributionexponentas a functionof non-dimensionalparameter p and fibre volume fraction. The thick dashedlinesketchesthe evolutionof a SiC/SiCyam duringthe process of successiverupturesof the fibres
Figure 6
352
This work has been performed within the framework of the BRITE-EURAM project BE-5462 with financial support from the European Commission, co-ordinated by Rolls-Royce plc (UK), in collaboration with SNECMA (France) and SEP (France). Financial support of CEIT by Rolls-Royce plc, SNECMA and SEP is also gratefully acknowledged. The funding received from the Spanish ‘Comisi6n Interministerial de Ciencia y Tecnologia (CICYT)’ and from the ‘Viceconsejeria de Educacibn, Universidades e Investigacibn’ of the Basque Government made possible the purchase of the nano-indentation equipment. A.M.D., M.R.E. and I.P. are grateful to the Commission of the European Union, Directorate General XII for Science, Research and Development, to the Spanish Ministry of Education and Science, and to the
Localized
Department of Education, Universities and Research of the Basque Government, respectively, for the grants received.
redistribution
9. 10. Il. 12.
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