An Cern So,85[611553-60(202) journal Failure of Crossply Ceramic-Matrix Composites Michael P. O Day and William A Curtin Division of Engineering, Brown University, Providence, Rhode Island 02912 The fast-fracture and stress-rupture of a crossply ceramic through the O plies, leading to through-thickness matrix cracks that natrix composite with a matrix through-crack are examined are bridged by fibers in the 0 plies, as shown in Fig. I(a). Bridging numerically to assess the importance of fiber architecture and only in the 0 plies leads to stress concentrations in the bridging the associated stress concentrations at the 0/90 ply interface on fibers near the 0/90 interface and failure of the composite is failure. Fiber bridging in the cracked 0 ply is modeled using a ultimately caused by the failure of the bridging fibers. The stresses line-spring bridging model that incorporates stochastic and in the bridging fibers of a crossply material have been determined time-dependent fiber fracture. A finite-element model is used by Xia et al. for elastically homogeneous materials, while earlier work focused on partially bridged cracks in unidirectional mate- presence of the bridged crack For both SiC/SiC and a typical rials. All of these works used the classic line-spring model and the oxide/oxide, the fast-fracture simulations show that as global bridging law of Marshall. Cox, and Evans, but with no fiber failure is approached, a significant fraction of fibers near the failure. Stress concentrations alone are also not sufficient for 0/90 interface are broken, greatly reducing the stress concen- predicting failure: i.e., the tensile strength is not the unidirectional tration. For fibers with low Weibull moduli (n 10), the strength divided by the maximum local stress concentration nal composite scaled by the appropriate fiber volume frac- ion, while for fibers with larger Weibull moduli (n 2 10). lessens the stress concentrations. The coupled phenomena of stress concentrations and fiber damage, and their influence on damage Stress-rupture simulations show that initially high stress con- ites with complex fiber architectures have not yet been studied there are modest (10-17%o) reductions in tensile strength and strength in crossplies, that pervade the mechanics of compe centrations are relieved as fibers fail with evolving time near the 0/90 interface and shed load away from the interface. For The majority of the literature has simply neglected the fiber a wide range of fiber properties, efficient load redistribution damage and proceeded to predict the tensile strength as if the occurs such that the crossply rupture lifetime is generally material were a unidirectional composite, In other words. if the within an order of magnitude of the unidirectional lifetime, strength of a unidirectional composite of fiber volume fractionis fracture strength. Overall, stress concentrations at the 0/90 on, then the tensile strength of a crossply or woven system of the interface are largely relieved with increasing load or time i9 y This result has proven accurate in the prediction of strength in same material has been estimated simply as (i/f)outs, where/, is to the nonlinear bridging response and preferential fiber fiber volume fraction in the direction of loading: typically f failure near the interface, resulting in crossplies that respond a number of different CMC systems. One major reason for the very similarly to unidirectional composites. success of the simple model is that, at the failure stress, there is typically a very high density of matrix cracks and, according to the results of Xia et al. the stress concentrations become small in 1. Introduction most cases. Not all composite systems have high crack densities T he behavior of unidirectional ceramic. near failure, however. Some systems also have high fiber/matrix com (CMCs) loaded in tension has bee lished experi nterfacial shear stresses, which cause higher stress concentrations ally and can be accurately predicted by g models 1.2 The important system of SiC/SiC can have both low crack densities and high interfacial shear stresses. In fact, such condi- strength, unidirectional composites are unsuitable for many appl ons tend to be optimal for design: low crack densities are usuall ations. This has led to the predominant use of crossply and woven coincident with high proportional limits so that materials can composites. The analytica results for unidirectional composites operate at reasonably high stresses with little or no damage,while that relate constitutive fiber, matrix, and interface properties to th high interfacial shear stresses lead to higher composite strengths tress-strain behavior and ultimate tensile strength (UTS)do not Under typical application situations of moderate stresses(well apply directly to crossply systems. Analytic models for fiber below the ultimate tensile strength), CMCs with matrix cracks bridging. which play an iniportant role in crossplies, are also not must also survive at high temperatures for long times. In this case generally applicable because the typical analyses (e.g. Marshall. Cox, and Evans, Danchaivijit and Shetty, and McCartney) are dependent. The reduction in stress-rupture lifetime of crossplies existing flaws or other processes that can be highly stress- ew methods of analysis ae needed. relative to unidirectional composites, due to architecture-induced The first damage mode in most crossply CMCs is matrix stress concentrations and accelerated fiber damage, has not cracking Matrix cracks typ cally start in the 90 plies and propagate been studied In this paper, we develop a coupled microscale/macroscale umerical model to examine both fast-fracture and stress-rupture in crossply CMCs. A finite-element ( FE) model is used to D, B. Marshall-comtributing editor determine the macroscale stress distributions in the presence of a matrix through-crack bridged by fibers in the 0 plies. Stochastic quasi-static and/or time-dependent fracture of the bridging fibers is then calculated based on the stresses obtained from the fe model ed July 2, 2001: approved April 1, 2002. and this microscale damage reduces the efficacy of the bridging hrough Grant No. F49620-99-1-0027, from the and leads to stress redistribution at the macroscale. Ultimately, the accumulated fiber damage near the 0/90 interface becomes large
Journal of the American Ceramic Sociery-O,Day and Curtin Vol. 85. No. 6 On =oco 0°py u=0 u=0 m crack 20 2 fiber atrIx bridging Fig. 1. (a) Representative section of periodic crossply composite, showing matrix crack ging in 0 plies; a unit cell is indicated by the dashed lines. (b) FE discretization of the unit cell, with boundary conditions and undeformed geometry enough to cause unstable propagation of fiber fracture across the 0 and Takeda, Under a far-field uniaxial applied stress o ding to global composite failur compare the brium leads to the relationship calculated fast-fracture strength to that estimated by the simple odels based on the unidirectional strength, and find generally ood agreement. This agreement stems from the fact that the 7=1-(1+2x T(x,1) ftening of the bridging due to fiber damage occurs preferentially at the 0/90 boundary where the stress concentrations are high 2l(x,) +q(T, leading to stress redistribution and a reduction in stress concen trations. The stress-rupture lifetimes of crossplies are compared with analytical and numerical determinations of unidirectional lifetimes and, at the same normalized applied stress, there is agreement within an order of magnitude for a wide range of fiber in the unbroken fibers at time I, g(T n) is the local fraction or parameters. The preferential and accelerated damage due to high robability of fiber failure at the fiber stress T(x n), T is the stresses at the 0/90 interface again acts to lessen the stress interfacial shear stress at the debonded/sliding fiber/matrix concentrations and deter global failure. We conclude that, in all cases, fiber damage largely relieves stress concentrations, resulting length determined from a shear-lag model, and 2 is the average matrix crack spacing. Although the unidirectional composite has II. the FE model, the fiber bridging model, and the fiber damage crossply system, where stress (and thus the fiber damage parameter is described. In Section Ill, we present fast-fracture and stress rupture results for SiC/SiC and oxide/oxide crossply composites. by taking 2- o to model a single matrix crack leading to and compare their behavior to unidirectional composites. In Section IV, we provide some further discussion and conclusions A) Fast-fracture: In fast-fracture. the time- independent fi IL. Model for Unidirectional and Crossply Composites ber damage q(T)arises from existing flaws in the fiber. Specifi We consider both unidirectional and o/90 crossply composite cally, a two-parameter Weibull model gives the probability of containing a matrix crack that extends completely through all failure in a length dz of fiber, over a stress increment o to g+ do matrix material perpendicular to the fiber axis. The crack is assumed to pass around all fibers in the 0 ply, leaving them intac s shown in Fig. I(a). Debonding along the fiber/matrix interface PAo,do, dz) is assumed to occur, with a residual interfacial sliding resistance T acting across the debonded interface region, The composites are intact fibo to contain a single matrix crack; the limitations of this where o is the characteristic fiber strength at a gauge length tion will be discussed in Section IV, During loading, the and m is the Weibull modulus. A critical fiber strength g. bers bridging the crack in the 0 plies will exert closure critical gauge length 8=ro /r can be identified for the problem. on the crack surface (1) Fracture of Unidirectional Composites In the unidirectional composite all fibers are parallel direction of applied loading. A general expression for mec where we expect to find one flaw of strength a. in a length 8. of equilibrium at the matrix crack plane has been derived by fiber. In the shear-lag fiber stress field around the matrix crack
June 2002 Failure of Crossply Ceramic-Matrix Composite 1555 maximum at the crack plane, and decreasing linearly until the Substitution of Eq(I1)into Eq.(10)and integrating yields an far-field stress is attained, the probability of fiber failure is o evolution equation for the flaw length. Using Eq.(12)the evolution of flaw strength S(z, /)is q(1=1-exp-m+1)fx)m(1+m"+)(5) s(D=1s(12-6(ryd (13) where a=fE Ee, with E, and Ee the fiber and overall composite Youngs moduli, respectively, and a tilde denotes a stress quantity normalized by ae. Using Eq.(2)the applied stress o can be where the initial flaw strength is S, (2)and the following normal- related to the fiber stress T(x) by izations have been introduced 0=fT(x)exp-1/(m+1)xym+(1+mm+)(6) T=tCo- (14a) Failure is the point at which no further increase in applied stress is C=5-1)AyK2 (14b) possible with increasing f ber stress 7(x),or The probability of failure in length dz of fiber, over the stress 0 increment o to o do is identical to Eg.(3), with the stress o replaced by the initial flaw strength S, (z). thus so that the UTS of the unidirectional composite with a single Pro, do, dz) dz ds (15) matrix crack is We simplify the fiber stress profile by neglecting any failure in the far-field. According to the shear-lag fiber stress profile, the fibe I +ma"+T stress decreases linearly with distance away from the matrix crack intil the far-field fiber stress level is reached. The fiber slip length To apply the unidirectional result (8) to a crossply composite, a I(n)=T(Or/2T is the distance over which the fiber stress would decrease to zero were it not interrupted by the far-field fiber stress scaling factor based on relative ply widths must be introduced. At The simplified fiber stress profile is the matrix crack in a cros sply composite, only the 0 plies carry load. Thus, for a crossply composite having respective ply widths of lo and loo the crossply tensile strength can be estimated gzn=T( (for I(r) lo which decreases linearly from the maximum value T(r) at the oP=o matrix crack to zero at a distance I, away from the matrix crack The evolution of this simplified fiber stress profile is shown in Fig 2. In this stress field, the probability of fiber failure anywhere along the slip length at any time up to the current time is elects any contribution of stress concentratio failure. In Section In(I), the predicted cross from Eq (9) will be compared with the FE results m l() (B) Stress-Rupture: Under stress-rupture conditions composite is subjected to a constant far-field stress while time evolves, resulting in increa sing fiber damage due to degradation of the fibers. Eventually, the composite will have damaged enough l (17) that the overall load level can no longer be maintained. The point stress-rupture lifetime at the given tensile ss the composite is the where q(T, i)is now explicitly a function of time. The stress- at which damage propagates unstably stress. In this case, E rupture lifetime of a unidirectional composite, at some constant (2)still holds, but the damage parameter g(T n) becomes time- dependent Here we assume that fiber degradation in time is governed by slow crack growth of existing flaws in the fiber. A Paris law describes the rate of crack growth as dr where a is the current crack length, K is the crack tip stress T(1 intensity factor, and B and A are the (possibly temperature dependent)crack growth exponent and rate constant, respectively. The stress intensity factor for tensile loading of a fiber is T K=ozn)yya where y is a geometric factor, and the fiber stress o. zn)is a function of both position z along the length of the fiber relative to the matrix crack at :=0 and time. The critical Mode I stress (t)z*t)(t2) intensity factor defines the strength S of a flaw of length a as Fig. 2. Evolution of shear-lag fiber stress profile with time around a matrix crack at
56 Journal of the American Ceramic Sociery-O Day and Curtin Vol. 85. No 6 remote stress, can then be found by solving the coupled Eqs. (I7) where m-fEr/(I- Em, and Em is the matrix modulus. We and (2). An iterative scheme is used to obtain a self-consistent include the effects of fiber failure, represented by the damage solution at a given time. Time is then incremented by some small parameter q as follows. We assume that in each small region of the mount. and a new solution is sought. The time at which no composite dx around point r there are a sufficient number of fibers converged solution exists corresponds to the stress-rupture lifetime such that the local response is identical to that of a unidirectional for a unidirectional composite composite. The local fiber stress 7(x) induces fiber damage q which evolves according to either Eq. (5)or Eq(17)for fast- (2) Finite-Element Mode fracture and stress-rupture simulations, respectively. T(r) is ob- The FE method is applied to simulate both fast-fracture and tained as a self-consistent outcome of the FE calculation. Fiber stress-rupture. The following discussion relates to the FE model in damage q due to the load T(r)then acts to weaken the bridging law general: aspects unique to each simulation type will be mentioned by reducing the fraction of fibers participating in the bridging as when appropriate, In formulating the macroscale mechanics prob- lem, a continuum approach is adopted. The discrete fibers and p(u,y)=(1-q)p(u) (19) matrix are replaced by a homogeneous, isotropic material with effective properties for each ply. The effective Youngs me ulus for the 0 ply corresponds to that for uniaxial loading of where I-q is the local probability of fiber survival. The single laminate along the fiber direction (longitudinal) and for dependence of the bridging law. p( q). on the local damage state the 90 ply corresponds to the transverse Youngs modulus of a q is explicit, and position dependence x and time dependence, if single laminate. The effective properties were determined from applicable, are implicit in u and q an Eshelby analysis, using the constituent properties given in An iterative solution procedure is used to obtain a self- Table I consistent solution in displacements, fiber stresses, and fiber Because of the periodic nature of the 0/90 laminate, a unit cell damage. A typical FE iteration proceeds as follows. The incre- geometry is appropriate. The unit cell and its FE discretization are mental displacement of the top edge is prescribed. For any iteration shown in Figs. I(a) and (b), respectively. The left and right edges (not yet converged)within this displacement increment, the of the unit cell correspond to the centers of the 0 and 90 plies. displacement field of the entire model was determined by the FE pectively. The left and right edges are constrained in the x solution from the previous iteration. The stress in intact fibers at direction and the z displacement of the top edge is prescribed. as the ith node (position x) bridging the matrix crack is calculated discussed below. The extent of the composite in the y direction is directly from the nodal displacements as T,=p(r )/f1-qi) assumed to be large enough such that plane strain conditions where 4, and u, were obtained in the previous iteration. Using T prevail. The model height h is taken to be 3(0 lou), which the probability of failure at the ith node is found from either Eq (5 1 100 bilinear plane strain quadrilateral elements comprised the FE or Eq.(17). in accordance with the type of simulation being performed. The line-spring stiffness contributions at the matrix crack are then k,=[dp(, q, )/du,lL p, with p(u. ) given by Eqs. (19) crack plane and the ply interface. All relevant geometric parame and(18). These are assembled into the global FE equations, which ters are given in Table L. are solved to provide the new displacement field. This procedure The prescribed displacement of the top edge is unique to the is repeated until the displacement solution converges. When the simulation being performed. In the solution has converged, the next incremental displacement displacement of the top edge is cally increased until composite failure occurs. In stress-rupture the goal is to maintain In fast-fracture simulations, evaluation of fiber damage by eq a constant level of remote stress, which is accomplished as (5) is clear. For stress-rupture Eq.(17)must be integrated follows. Over a very small time the composite is loaded to the numerically, as follows. At some time i and at every position x,the ply moduli, as well as the bridging stress variation along the matrix the length of the fiber, up to the current slip length /.().Evaluation of the first term inside the brackets is straightforward, but the constant at the desired level. but the FE model is loaded through nature of the time-varying slip length 1, (r) in the second term prescribed displacements. In general, as damage accumulates, a requires that the lower limit of the time integral be modified greater incremental displacement is required to maintain the Within the present assumption of no far-field fiber stress, the slip constant remote stress level. In the FE procedure, the current length and stress profile evolution along any fiber is shown in Fig remote stress and remote stress history are used to predict the 2, where t-0 represents the matrix crack. When the position ecessary incremental displacement to satisfy the constant remote integral in Eq (17) is evaluated at some z", the time integral is stress condition. Fluctuations from the desired stress level are on physically meaningful only at times I'such that sI(). From the order of 00I% Fig. 2, we see that the stress at z is zero until the slip length The effect of fiber bridging at the matrix crack plane is creases such that z=I(1). This means that for any time less devised by Danchaivijit and Shetty"(DS), which incorporates the Computationally, the lower limit of the time integral in Eq(17 physically correct fiber stresses in the uncracked matrix material. thus becomes i with no loss in generality In the absence of any fiber damage and uniform remote loading. Several checks were used to verify the FE model, The FE the Ds bridging law relates the closure traction p(un) to the crack fast-fracture strengths of unidirectional composites were compared opening displacement 2n(r) according to with the analytical UTS Eq (8), and were in exact agreement. The 16(1+n)Efr(x) FE stress-rupture lifetime of a unidirectional com WEIS compared with the analytical lifetime, obtained by solving the coupled Eqs.(17)and (2), and again the FE results were in exact (18) agreement. Table 1. FE Input Material and Geometric Parameters Er(GPal En, (GPal 。fGPa) T(MPa lo (um) SiC/SiC 269 0.16 310 0.342 7.5 Oxide/Oxide 372 0.248
June 2002 Failure of Crossply Ceramic-Matrix Composites 1557 Il. Results FE simulations were performed for a SiC/SiC composite and a typical oxide/oxide. The overall trends observed were similar for the two systems; thus we present fast-fracture results for the 0.8 SiC/SiC system and stress rupture results for the oxide/oxide. 00. I) Fast-Fracture Results The normalized continuum bridging stress p(u, q)outs and fiber 日xb=0. damage parameter g are shown for the SiC/SiC composite in Figs. 3(a)and(b), respectively, at various levels of applied loading. At loads less than 50% of the crossply UTS. a relatively large stress x/L=1 concentration(approaching 2)exists but a negligible percentage of 2 fibers have fractured. With increasing load, fiber damage accumu- lates more rapidly, particularly near the 0/90 interface, reducing the stress concentration. At an applied stress of% of the rossply UTS. the region of the 0 ply at the 0/90 interface just reaches the limiting value cf the unidirectional UTS out. Since no u region of the 0 ply can exceed ous, as the applied stress is ncreased the near-interface region sheds load to fibers away from Fig. 4. Bridging law history at four points along the 0 ply ( SiC/SiC, m he interface. Simultaneously, with increasing strain, fibers in the 5); symbol denotes global crossply failure and stress is normalized by g near-interface region are .damaging so rapidly that this region ctually supports a decreasing amount of stress. Figure 3(a) show that, at failure, the stress in the highly damaged near-interface MCE-type law with P(u)=Au, where A is a constant indepen- region is significantly belo w the unidirectional failure stress. The dent of remote stress, the curves in Figure 4 collapse to a single evolving damage softens the near-interface region significantly curve and the bridging law of Eq. (19)is unique nd decreases the stress concentration as the applied stress is The FE determination of the UTS is compared with the simple increased further. The net result is an almost complete elimination analytical prediction, Eq (9), for both composite systems in Table of the stress concentration, and hence the tensile strength is well Il. We have considered a balanced crossply (loo =l,)and loo- approximated by the scaled unidirectional theory, as discussed Lo, which can roughly approximate current woven materials with further below the longitudinal tows represented by the 0 ply and the transverse It is useful to examine the local bridging law at several locations tows and matrix-rich regions roughly represented by the wider 90 along the 0 ply. The DS bridging law, Eq.(18), describes p(u)as ly. At lower Weibull moduli (m= 5) the scaled unidirectional a monotonic function of u, but the bridging is not monotonic when heory slightly overpredicts the UTS of the crossply SiC/SiC fiber damage occurs With clamage, the bridging law has hardening system. For larger values of Weibull modulus (m= 20). the shows the bridging history of four points along the 0 ply, up to oxide/oxide system (not shown) shows slightly smaller stress composite failure. As implied by the failure stress distribution of concentrations and greater ability to relieve them by 3(a). the most heav ly damaged regions evolve on the Thus the oxide/oxide UrS follows the trends found for oftening portion of the bridging law. Globally, the composite is but the differences with the scaled unidirectional pred stable even though local re zions of softening exist. Figure 4 also even smalle shows that all points of the 0 ply basically follow the same The variation of the Weibull was performed at a fixed underlying bridging law; it is the degree of local stress concentra- value of o.(see Table D). As the Is increases, the tion and fiber damage that determines how much bridging actually stress range over which fiber se arrows. In the occurs. The slightly different bridging described in Fig. 4 is solely limiting case of an infinite Weibull modulus, failure becomes due to the o dependence of the DS bridging law. Indeed for an extremely brittle and occurs at the instant any point along the 0 ply 99% 80 10 104 (b) vior for balanced (o=loo )SiC/SiC composite, = 5: (a)norma bridging stress versus distance a parameter versus distance along O ply matrix crack. Quantities are shown at various percentages of the applied stress to event u时