SP 190g site Mater. Vol 8 No. 1. pp 97-106(1999) Adv. Comp Probabilistic-statistical approach to the mechanical behavior of ceramic matrix composites(CMCs) J. LAMON Laboratory for Thermostructural Composites, UMR-/(CNRS-SEP/SNECMA-UBI) University of bordeaux 1, 3 Allee de La boetie, 33600 Pessac, france Abstract-The approach to the mechanical behavior of CMCs that is proposed, is based on a statistical-probabilistic description of multiple matrix cracking and fiber failures. It is applied to unidirectional and woven SiC/SiC composites. The predicted stress-strain behaviors were in good greement with available experimental data. The influence of various factors including porosity. loading conditions and constituent properties is anticipated. Keywords: Ceramic matrix composite: mechanical behavior: damage; fracture probability: failure: tatistics I INTRODUCTION Ceramic matrix composites combine brittle materials including the fiber and the matrix. Nevertheless they are capable of a highly non-linear stress-strain behavior reflecting damage tolerance. Damage involves multiple microcracks or cracks that form in the matrix and then are arrested by the fibers. The ultimate failure occurs when a critical number of fibers have failed. Matrix cracking and fiber failures are brittle failure phenomena. They are initiated by microstructural heterogeneities distributed randomly in the matrix and in the fibers Fracture-statistics based approaches are appropriate for modeling the defect- induced failures of brittle materials. Several approaches to brittle fracture have been proposed in the literature. The Weibull model provides a satisfactory approximation in the presence of uniaxial stress states, More fundamental approaches such as the Multiaxial Elemental Strength Model are required for complex shapes subject to multiaxial stress states [1]. Damage and failure models based upon statistical approaches to brittle failure have been developed to predict the ultimate strength and the stress-strain behavior [6, 7] of unidirectionally reinforced CMCs subject to tensile loads, and the scenario of matrix damage in 2D woven SiC/SiC composites [81
Lamon The paper presents damage and failure models based upon statistical approaches to brittle failure to predict the stress-strain behavior of composites at various scales including microcomposites (elementary scale)[]. minicomposites(intermediate scale)[7I and 2D woven composites [8]. A microcomposite consists of a concen tric cylinder element containing a single fiber with a coating (i.e. interphase) plus a matrix annulus. This specimen is appropriate for designing interphases 191. Micro composites are not examined in the paper. A minicomposite is a bundle of parallel fibers coated with a layer of interfacial material and a matrix. Minicomposites are representative of the matrix infiltrated bundles in textile ceramic matrix composites reinforced with fabrics of woven bundles 2. APPROACHES TO MATRIX CRACKING AND ULTIMATE FAILURE IN MINICOMPOSITES 2 / Matrix cracking As matrix cracking proceeds in a minicomposite specimen loaded in tension parallel to the fiber axis, the matrix becomes subdivided into smaller and smaller fragments A fragment is defined as the uncracked element bounded by two transverse matrix cracks(Fig. 1) Matrix cracking is modelled as the brittle failure of uncracked fragments. This approach differs from the conventional one developed for fiber fragmentation which considers the whole cracked volume. The probability of failure of a fragment is obtained from integration of the 2-parameter Weibull equation, for the specific stressfield operating on the matrix In the presence of n matrix cracks, the matrix is subdivided into(n+ 1)fragments (Fig. I). The probability of brittle failure of the ith matrix fragment of length 2l; is given by the following equation PMi=1-exp-2SM M where SM is the cross-sectional area: mM and ooM are the statistical parameters pertinent to the matrix: OM represent the stresses acting on the matrix Initial flaw Matrix crack 212 Figure 1. Schematic diagram showing the formation of matrix fragments
Probubilistic-statistical approach to behavior of CMCs Debonding caused by deflection of the matrix cracks into the interphase affects ocally the applied stress-field, inducing stress gradients in the matrix adjacent to the debond. Incorporating the appropriate equations for oM, and performing integration of equation() gives exp-2SM\OoM VM(1+a) (mM+B),(2) where ldi is the debond length in this ith matrix fragment Matrix multiple cracking was then modelled as follows. A given fragment fails when its strength becomes smaller than the stress acting on the matrix. The fragment strengths depend on the fragment sizes. Fragment size is dictated by the location of failure at a previous step. It is derived from the probability of location of the critical flaw. The fragment strength(oM) derives from the probability of fragment failure. lai m+By/mM l1m+1 where or is a reference matrix strength for the volume VoM of matrix. V;is the fragment volume, lai is the debond length, 2l; is the fragment length, B= lo/ldi where lo is a reference distance along which the Poisson effect is operative 2.2. Fracture of fibers The fracture of fibers involves the following features seen on SiC/SiC and C/Sic composites and minicomposites during tensile tests: (i)the individual fiber breaks occur at high loads near ultimate failure; and (ii) the fibers fail only once After saturation, the crack spacing distance is generally a few microns scale whereas the debond extends along the entire fibers. The load carrying capabilit of the matrix is tremendously reduced or annihilated. Once a fiber has broken anywhere in the gauge length of the specimen, it is no longer capable of carrying he load 4, 5 Probability of failure for the fibers is determined by the stress state induced by the matrix cracks and also by the law of load sharing among the surviving fibers. as fibers fail individually. a global load sharing was assumed as observed in fiber bundles. The failure probability was expressed in terms of an equivalent fiber length (denoted Lequi )defined as the length of a fiber subjected uniformly to the peak stress omax for the same failure probability. Failure probability for a fiber within he minicomposites is given by the following equation p
lamon where F is the applied force, Ar is the cross-sectional area of a single fiber, A' is the total cross sectional area of all the fibers present in the minicomposites, mr and oor are the statistical parameters pertinent to the fibers, and a is the fraction of individual fiber breaks. a represents the failure probability of the Nth fiber. The number of fibers N that are broken under a given force F is related to the initial number of fibers No by the following equation N= NoP. Inserting the expression for P into equation (6) the fraction of surviving fibers is given by q=1-0=exp- Lequi(a) A(I-a)oor The ultimate failure results from an instability in the evolution of individual fiber failures. It is characterized by the following criterion 8F Sa 2.3. Prediction of the mechanical behavior of minicomposites Minicomposite deformations are dictated by the fibers. They were derived from the stress field operating on the fibers, as a function of the number of matrix cracks and the fraction of broken fibers(a). The properties required for the computations are given in Table I The force-strain curves predicted for the SiC/SiC minicomposites compared fairly well with the experimental data(Fig. 2). The model allowed determination Mean characteristics of the SiC/SiC mini v(%) 31.5 Fiber Youngs modulus(GPa) Matrix Youngs modulus(GPa) Statistical parameters matrIx: IM 608(1.66) matrix: GoM(Vo= I m)(MPa) 10.51(1.23) fiber; mt 5.45 fiber:oor(Vo=I m)(MPa) r(MPa) 115 hysteresis loops
Prmobabhilistic-statistical approach to behavior of CMCs Deformation (e) Figure 2. Comparison of the predicted and experimental mechanical behaviors for a SiC/SiC of the influence of constituent properties on the response of minicomposites. It was shown that the matrix influence is similar to that of the interfacial shear stress Low stresses and a plateau-like domain are enhanced by a matrix displaying a low strength, a high Weibull modulus and a high Young modulus. Variability in fiber strength causes a significant scatter in the minicomposite strain-to-failure. Finally, it was shown that the minicomposites are significantly sensitive to size effects 3. PREDICTION OF DAMAGE EVOLUTION IN A 2D WOVEN COMPOSITE In the composites reinforced with fabrics of fiber bundles the matrix damage is influenced by the microstructure [10]. The 2D SiC/SiC made by Chemical Vapor Infiltration(CvI) display a highly heterogeneous microstructure consisting of woven infiltrated tows that behave as physical entities, large pores(referred to as macropores)located between the plies or at yarn intersections within the plies, and a uniform layer of matrix over the fiber preform (referred to as the intertow matrix) Much smaller pores are also present within the tows. Extensive inspection of the composite under a tensile load using a microscope has shown that matrix cracking affects first the intertow matrix, then the transverse infiltrated tows and finally the longitudinal infiltrated tows [101 3. 1. Determination of failure probabilities The matrix damage evolution in a 2D SiC/SiC composite was predicted using a finite element analysis of failure probabilities(Fig. 3)18. 11. First, the stresses in the matrix are computed using a finite element code (MARC produced by MArC analysis). Then, failure probabilities are determined using the FLAG post processor 8, 11]. FLAG uses the finite-element output (principal stresses), and a material data
J. lamon Face I&tE Face 2 Figure 3. Cell and finite element mesh used in the analysis file including the statistical parameters pertinent to composite basic constituents The FLAG program includes the multiaxial elemental strength model [1, 12] for handling multiaxial fracture statistics. The multiaxial elemental strength model was developed to predict the failure of ceramic materials under various stress-states and geometries. It is based upon the premise that the pre-existing flaws in the material can be characterized by their flaw extension stress, or strength S(referred to as the elemental strength), The flaw distributions are thus described by the distribution in elemental strengths S. The failure probability is then derived from the distributio in s The failure probability of a volume element of matrix (i.e. a single finite element mesh unit) is calculated using the following equation oov/ Iv(mv. on (9) where Py is failure probability, o, o2 and o3 are the principal stresses(or> 02>03), my and ov are the flaw strength parameters, V is the volume of the considered matrix element. The function Iv(.) is detailed elsewhere [1 It depends upon the local principal stress field through a non-coplanar strain energy elease rate criterion for crack extension [1, 121 3.2. Principle of matrix cracking simulation The applied deformation(or the applied stress) is increased incrementally. At each increment, the stress-state and the unit mesh failure probabilities are computed. A crack in the intertow matrix or in the matrix in the transverse tows(perpendicular to the loading direction), or a damaged zone in the matrix in the longitudinal (parallel to the loading direction) is introduced at the location of the max unit mesh failure probability when this computed failure probability reaches value of 1, in order to represent definite events. The cracks are introduced in the mesh by iteratively splitting the nodes. The damaged zones are characterized by an effective Youngs modulus according to the density of matrix cracks, in order
Probabilistic-statistical approach to behavior of CMC to take into account the contribution of the debonded fibers in the deformations of these longitudinal tows 3.3. Application to a 2D SiC/SiC composite The properties of constituents required for computations are summarized in Tables 2 and 3 A cell of 2D SiC/SiC composite under uniaxial tension was examined first The damage evolution predicted was found to be in excellent agreement with that identified on practical 2D woven SiC/SiC composites under a uniaxial tensile load (Table 4). The predicted stress-strain curves and Youngs modulus compared satisfactorily with experimental data( Fig. 4) However, a certain discrepancy was observed which was attributed primarily to the matrix /tow interactions Then, this approach was applied to predict the damage evolution and the stress- strain behavior of a fully dense 2D woven SiC/SiC composite. The results indicated that the macropores have a beneficial effect on the mechanical behavior(Fig. 5). The short macropore-induced cracks are less detrimental than the long transverse cracks that would appear in the fully dense SiC/SiC Finally, the potential of this approach to handle complex loading conditions was illustrated on a cell subjected to non-uniform forces. The approach predicted the damage evolution(involving intense debonding) observed on practical CMCS Table 2 Main mechanical properties of composite constituents onstituents Youngs modulus( GPa) Isson s ratIo Fiber 200 0.25 Matrix 400 Longitudinal tows 0.20 ongitudinal tow Transverse Statistical parameters of the SiC matrix in the different constituents of the SiC/SiC composite Constituents Weibull modulus Scale factor(MPa macrocomposite Incomposite Interply matrix Transverse tow Longitudinal tow matrix
104 Lamon Table 4 Comparison of the predicted evolution of matrix damage with experimental results in a 2D SiC/SiC composite under uniaxial tension Prediction Experiment deformation age deformation observed dam (%) 0.04 crack initiation 0.035 crack initiation 0.085 ransverse tows transverse tows ng In 0.185 cracking in longitudinal longitudinal towS Stesses (MPa) I 0.1125 0.225 0.375 deformations (56) Y modulus (GPa 50 0.1125 0.3375 deformations (%) Figure 4. Comparison of experimental (1)and predicted (2). (3)stress-strain behaviors and Youngs moduli: (2)[131(3)present analysi
Probabilistie-statistical approach to behavior of CMC 105 Stresses (MPa) 0.11250.2250.3375045 deformation (%o) Young's oulu (GPa) I 0.1125 0.225 0.3375 Figure 5. Comparison of the stress-strain behaviors and the Youngs moduli predicted (1) for the onventional 2D SiC/SiC composite, and (2) for the corresponding fully dense 2D SiC/SiC under bending conditions. The stress-strain curve exhibited the features of the mechanical behavior evidenced on the practical 2D woven SiC/SiC composite 4. CONCLUSIONS The model of the force-strain behavior of minicomposites which was presented is based on a probabilistic-statistical description of matrix cracking and fiber bundle failure. The minicomposites are assimilated to fiber bundles subjected to a typical stress-state induced by the presence of matrix cracks and associated debonding. A Weibull type statistical-probabilistic model was developed for the description of the matrix cracking process and fiber failure. This model provided stress-probability equations as a function of the number of matrix cracks and various constituent The force-strain behaviors that were predicted from the characteristics of con stituents were in excellent agreement with the experimental ones, thus validating the model. Important trends in the influence of constituents properties on the me- chanical behavior of minicomposites were anticipated
J, lamon The damage evolution, which was predicted from properties of basic constituents using failure probability computations, was found to be in excellent agreement with that identified on practical 2D woven SiC/SiC composites. The predicted stress-strain curves and Youngs modulus compared satisfactorily with experimen- tal data. However a certain discrepancy was observed which was attributed primar- ily to the matrix/tow interactions Then, this approach was applied to a fully dense 2D woven SiC/SiC composite The results indicated that the macropores have a beneficial effect on the mechanical behavior. Finally, the potential of this approach to handle complex loading conditions was illustrated on a cell subjected to non-uniform forces. The results indicate that this approach may be used to simulate matrix damage and related stress-strain behavior of CMCs from properties of basic constituents Acknowledgements This work has been supported by the Ministry of Education and Research. CNRS and SEP. The author acknowledges the contribution of former students including L. Guillaumat N. Lissart, B. Thommeret and Ch. Percevault REFERENCES I, J. Lamon and A. G. Evans. J. Am. Ceram, Soc. 66. 177(1983). 2. w.A. Curtin, J. Am. Ceram Soc. 74 2837(1991) 3. P S Steif and H R. Schwietert, Ceram. Eng. Sci. Proc. 11, 1567(1990) 4. M. Sutcu, Acta Metall. 37. 2567(1989) 5. H Cao and M. D. Thouless. J. Am. Ceram. Soc. 73, 2091(1990) 6. L Guillaumat and J, Lamon Int, J. Fract. 82. 297(1996) 7. N. Lissart and J. Lamon, Acta Metall. 45. 1025(1997) 8. J. Lamon, B. Thommeret and C, Percevault J. Eur: Ceram. Soc. 18, 1197(1998) 9. J. Lamon N. Lissart C. Rechigniac, D. M. Roach and J M. Jouin, Cera. Eng. Sci. Proc. 1115 1993) 10. L Guillaumat and J. Lamon, Revue des Composites et des Materianr Avances 3. 159(1993) I1, J. Lamon and B. Thommeret, Ceran. Trans. 57. 77(1995) 12. J. Lamon. J Am Ceram Soc. 73 2204(1990) 13. L. Guillaumat and J. L amon, Compos. Sci. Technol. 56. 803(1996)