August 1997 Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites 2045 tress. The effect of roughness on interfacial friction is consid where ered and discussed first; this is followed by a presentation of the calculated effects on debond length, either as a function of (1+x) fiber stress or at fiber fracture. Interactions between rough f1 x(1+x) 2 nd other variables that have been examined by a parametric evaluation are included in the discussion. Next, the validity of g tg 1(1-x) the constant shear stress(T) approximation is examined by (4) for t g the possibility of extracting and using a single value for T to predict debond lengths as a function of fiber stress Finally, the calculated fiber stress-debond length relationship was used in the model of Curtin, 32 to predict composite stress- 2 strain behavior and ultimate strengths in the presence of inter- facial roughness. The results are compared with predictions 十 -1(1-x)′ hat have been obtained using the same approach in a conven tional manner using a constant shear stress T with values that would be derived from conventional methods such as the fiber ='=n/ pushout or pullout tests (7) The case of an interface crack propagating away from a matrix crack is shown schematically in Fig. 2. The elastic misfit is assumed to cause uniform radial strains in the fiber and the matrix, summing to the magnitude of the misfit. As 2μ'b1 discussed above, the debonded length comprises two different (9) egions of misfit behavior In region Ill, the fiber has slid past the matrix by a distance greater than the half-period, d. The 2uL'Be roughness is assumed to be nonaxisymmetric, and there is no d’=tan (10) position of fit between the fiber and the matrix, except in the original position; hence, there is no further change in misti ith additional sliding. The effects of roughness in this large a=y+ocn+σ (11) displacement region can be modeled as suggested in earlier works, 18,25 with u as the friction coefficient and the quantity Ge AaAT+(/r) as a constant mismatch strain(no effect of Re sliding distance on clamping stress), where AaAT is the con- tribution from the thermal mismatch. Under these conditions (13) region II will have a constant length, but moves with the debond crack tip, whereas region IlI will increase in length In the above equations, B. ar, bu, and Eh are functions of the the debond crack progresses elastic properties of the matrix and fiber, as well as the fiber Within region Il, the roughness-induced clamping stress is volume fraction, as defined by Hutchinson and Jensen. G is been described elsewhere that derives expressions that relate residual axial stress in the fiber, and om, a residual stress the length of the debond crack as a function of the fiber stress rameter whose magnitude gives the fiber stress at which the The model includes the effects of the thermal mismatch, the Poisson's contraction of the fiber exactly cancels the residua Poisson's contraction of the fiber. the axial residual strain in clamping stress during fiber pullout. When the roughness am- litude is zero, x l and the equations then reduce to those computational tractability is such that the misfit increases lin- same equations were used, with x=0 and the radial clampino than the half-period d and is constant thereafter. Approxima- +(h/R). The specifics of the calculations are discussed in more tions in the model require that the roughness amplitude h and detail in the Appendix od d be much smaller than the fiber rad Using the approach above, the debond length was calculated, result of the model is that, over region Il, an effective friction as a function of fiber stress, for different sets of constituent coefficient, u, replaces the friction coefficient, u, between the parameters. For all the calculations, the boundary conditions fiber and matrix. The effective friction coefficient is given by that corresponded to that of a multifiber pullout were used(no radial displacement at the matrix surface; type ll as defined by u+ tan 6 Hutchinson and Jensen), because this corresponds to the ten- (1) sile testing of a composite. Because of the lack of reliable data on the Poisson's ratio of the fibers, and to simplify the calcu- where tan 0= h/d for the sawtooth roughness shape that is lations. the fibers and matrices were assumed to have the same assumed in the model with the requirement that tan 0< 1/u Poissons ratio. The debond length at a fiber stress that equals The roughness-induced clamping stress at any point increase he fiber strength was considered to be the parameter of most linearly with increasing relative displacement between the fiber interest. It must be noted that this debond length at the fiber and the matrix. The final equations relate the debond length, L, fracture stress is not the same as the mean pullout length that to the remote fiber stress, Oa, and fiber displacement, u, in is measured at the composite fracture surface, however, the two terms of the properties of the fiber, matrix, and interface. These re closely related through a prefactor that is dependent on the relations can be described in terms of two functions, f, and f2 Weibull modulus of the fiber strength. I It also is noteworthy that the fiber stress at fiber fracture is not the same as that ( (2) which is measured in a fiber test that usually has a gauge length of 1 in. the effective gauge length for the fiber failure within a composite can be much lower. The best estimates of fiber fracture stress in composites may be those which are obtainestress. The effect of roughness on interfacial friction is consid￾ered and discussed first; this is followed by a presentation of the calculated effects on debond length, either as a function of fiber stress or at fiber fracture. Interactions between roughness and other variables that have been examined by a parametric evaluation are included in the discussion. Next, the validity of the constant shear stress (
) approximation is examined by studying the possibility of extracting and using a single value for
to predict debond lengths as a function of fiber stress. Finally, the calculated fiber stress–debond length relationship was used in the model of Curtin1,32 to predict composite stress– strain behavior and ultimate strengths in the presence of inter￾facial roughness. The results are compared with predictions that have been obtained using the same approach in a conven￾tional manner using a constant shear stress
with values that would be derived from conventional methods such as the fiber pushout or pullout tests. II. Approach The case of an interface crack propagating away from a matrix crack is shown schematically in Fig. 2. The elastic misfit is assumed to cause uniform radial strains in the fiber and the matrix, summing to the magnitude of the misfit. As discussed above, the debonded length comprises two different regions of misfit behavior. In region III, the fiber has slid past the matrix by a distance greater than the half-period, d. The roughness is assumed to be nonaxisymmetric, and there is no position of fit between the fiber and the matrix, except in the original position; hence, there is no further change in misfit with additional sliding. The effects of roughness in this large￾displacement region can be modeled as suggested in earlier works,18,25 with as the friction coefficient and the quantity T + (h/Rf ) as a constant mismatch strain (no effect of sliding distance on clamping stress), where T is the con￾tribution from the thermal mismatch. Under these conditions, region II will have a constant length, but moves with the debond crack tip, whereas region III will increase in length as the debond crack progresses. Within region II, the roughness-induced clamping stress is not constant. This case has been treated in the model that has been described elsewhere31 that derives expressions that relate the length of the debond crack as a function of the fiber stress. The model includes the effects of the thermal mismatch, the Poisson’s contraction of the fiber, the axial residual strain in the fiber, and the roughness at the fiber/matrix interface. The simple form of roughness to which the analysis is restricted by computational tractability is such that the misfit increases lin￾early as the displacement increases for displacements of less than the half-period d and is constant thereafter. Approxima￾tions in the model require that the roughness amplitude h and half-period d be much smaller than the fiber radius. A key result of the model is that, over region II, an effective friction coefficient,
, replaces the friction coefficient, , between the fiber and matrix. The effective friction coefficient is given by
= + tan  1 − tan  (1) where tan  h/d for the sawtooth roughness shape that is assumed in the model with the requirement that tan  < 1/. The roughness-induced clamping stress at any point increases linearly with increasing relative displacement between the fiber and the matrix. The final equations relate the debond length, l, to the remote fiber stress, a, and fiber displacement, u, in terms of the properties of the fiber, matrix, and interface. These relations can be described in terms of two functions, f1 and f2: u =
 Eb f1 (2) a − f + = f2 (3) where f1 = 4
g − 1 1 − x 2 +
g − gx − 2 x
1 + x exp −
1 + xz
2  −
g + gx − 2 x
1 − x exp −1
1 − xz
2  (4) f2 = −
g − gx − 2 2x exp −
1 + xz
2  +
g + gx − 2 2x exp −1
1 − xz
2  (5) z
= l (6) x =
1 − 4
2 12 (7) g =  (8)  = 2
b1 Rf (9) 
= tan 
2
BEm EbRf 2 (10)  = + fo + + Ro (11) = 2
EbGc Rf 12 (12) f + = a1fa + fo + (13) In the above equations, B, a1, b1, and Eb are functions of the elastic properties of the matrix and fiber, as well as the fiber volume fraction, as defined by Hutchinson and Jensen.33 Gc is the interface toughness, f the fiber volume fraction, fo + the residual axial stress in the fiber, and Ro a residual stress pa￾rameter whose magnitude gives the fiber stress at which the Poisson’s contraction of the fiber exactly cancels the residual clamping stress during fiber pullout. When the roughness am￾plitude is zero, x 1 and the equations then reduce to those derived earlier25,33,34 for smooth fibers. Thus, for region III, the same equations were used, with x 0 and the radial clamping stress set to that which results from a mismatch strain of T + (h/Rf ). The specifics of the calculations are discussed in more detail in the Appendix. Using the approach above, the debond length was calculated, as a function of fiber stress, for different sets of constituent parameters. For all the calculations, the boundary conditions that corresponded to that of a multifiber pullout were used (no radial displacement at the matrix surface; type II as defined by Hutchinson and Jensen33), because this corresponds to the ten￾sile testing of a composite. Because of the lack of reliable data on the Poisson’s ratio of the fibers, and to simplify the calcu￾lations, the fibers and matrices were assumed to have the same Poisson’s ratio. The debond length at a fiber stress that equals the fiber strength was considered to be the parameter of most interest. It must be noted that this debond length at the fiber fracture stress is not the same as the mean pullout length that is measured at the composite fracture surface; however, the two are closely related through a prefactor that is dependent on the Weibull modulus of the fiber strength.1 It also is noteworthy that the fiber stress at fiber fracture is not the same as that which is measured in a fiber test that usually has a gauge length of 1 in.; the effective gauge length for the fiber failure within a composite can be much lower. The best estimates of fiber fracture stress in composites may be those which are obtained August 1997 Predicted Effects of Interfacial Roughness on the Behavior of Selected Ceramic Composites 2045