Y-F. Liu et al./ Mechanics of Materials 29(1998)111-121 2. 2. Bridging law in consideration of interfacial verified that the energy release rate showed qualita- friction and debonding toughness (Yuuki and Liu, tive agreement with finite element analysis results in 1994) determining debond length and predicted results agreeing well with previous experimental data, al An axisymmetrical fiber-matrix model having a though singular stress and strain fields at the debond debond length of I and a constant sliding stress, T crack tip were simplified in deriving the energy between 0<:<I is used(Fig. 2). R is the fiber release rate for the debond crack(Yuuki and Liu, radius and Rm is the matrix radius, and fiber volume 1994). This energy release rate expression is a sec- fraction,f, is defined as f=(R/Rm). The outer ond-order function of the debond length and will be boundary conditions of the cylindrical model are set outlined below to stress-free. A uniform tensile stress, o, is applied Extruding bridging fiber and surrounding matrix to the upper end of the model. With the traditional as an axisymmetrical cylinder, the crack opening shear-lag method and energy balance arguments, we displacement, u, is expressed as(Yuuki and Li have derived an all-inclusive energy release rate, G 1994) by using a Lame solution from Hutchinson and Jensen(1990)after considering an infinitesimal ad vance of the debond crack( Gao et al., 1988 Sigl and b,F 2(1) Evans. 1989: Hutchinson and Jensen. 1990: Yuuki and Liu, 1994). Influences of various important pa- where ef=50(ar-am)dr. Here, AT is the tem- rameters on the energy release rate were demon- perature change from bonding, a's and b, are mate strated and its physical significance clarified. It wa rial- and geometry-relevant parameters given in Hutchinson and Jensen(1990). Other parameters are explained in Fig. 2 Eq.(1) is a nonlinear function of the debond Matrix length, 1, which needs to be determined. Assuming that the applied stress, o, is fixed, the debond length Material Properties upon the applied stress is obtained from the follow- Fiber: Er, Vr,af, af ing condition Matrix: Em, Vm. am G.≥G where Gic is the debonding toughness obtained ex- perimentally and G is normalized as Z▲ G G E R e a,BE Roof 向1a2Ene 2f ic cylindrical model: (a)longitudinal section, (b)transverse section. Also shown are material and geometric where a's and b's are again material- and definitions. Material properties are indicated in this figure, where geometry-relevant parameters given in Hutchinson E and v are Youngs moduli and Poissons ratios, respectively, and subscripts, f and m, refer to fiber and matrix, respectively; ar and Jensen(1990). Within the two roots of Eq (2) is thermal expansion coefficient and subscripts, r and z, for a only the small one is physically meaningful and dicate radial and axial directions, respectively should be taken as the critical debond length whenY.-F. Liu et al.rMechanics of Materials 29 1998 111–121 ( ) 113 2.2. Bridging law in consideration of interfacial friction and debonding toughness Yuuki and Liu, ( 1994) An axisymmetrical fiber–matrix model having a debond length of l and a constant sliding stress, t , between 0FzFl is used Fig. 2 . Ž . R is the fiber f radius and Rm is the matrix radius, and fiber volume Ž .2 fraction, f, is defined as fs R rR . The outer f m boundary conditions of the cylindrical model are set to stress-free. A uniform tensile stress, s , is applied to the upper end of the model. With the traditional shear–lag method and energy balance arguments, we have derived an all-inclusive energy release rate, G ,i by using a Lame solution from Hutchinson and ´ Jensen 1990 after considering an infinitesimal ad- Ž . vance of the debond crack Gao et al., 1988; Sigl and Ž Evans, 1989; Hutchinson and Jensen, 1990; Yuuki and Liu, 1994 . Influences of various important pa- . rameters on the energy release rate were demon￾strated and its physical significance clarified. It was Fig. 2. Axisymmetric cylindrical model: a longitudinal section, Ž . Ž . b transverse section. Also shown are material and geometric definitions. Material properties are indicated in this figure, where E and n are Young’s moduli and Poisson’s ratios, respectively, and subscripts, f and m, refer to fiber and matrix, respectively; a is thermal expansion coefficient and subscripts, r and z, for a indicate radial and axial directions, respectively. verified that the energy release rate showed qualita￾tive agreement with finite element analysis results in determining debond length and predicted results agreeing well with previous experimental data, al￾though singular stress and strain fields at the debond crack tip were simplified in deriving the energy release rate for the debond crack Yuuki and Liu, Ž 1994 . This energy release rate expression is a sec- . ond-order function of the debond length and will be outlined below. Extruding bridging fiber and surrounding matrix as an axisymmetrical cylinder, the crack opening displacement, u, is expressed as Yuuki and Liu, Ž 1994 :. 2 1 1 s t l T us ya lq qa ´ l , 1Ž . 1 2z ž / b F E ER 2 m mf T DT Ž f m where ´ sH a ya .d . Here, DT is the tem- z0z T perature change from bonding, a’s and b are mate- 2 rial- and geometry-relevant parameters given in Hutchinson and Jensen 1990 . Other parameters are Ž . explained in Fig. 2. Eq. 1 is a nonlinear function of the debond Ž . length, l, which needs to be determined. Assuming that the applied stress, s , is fixed, the debond length upon the applied stress is obtained from the follow￾ing condition: G GG , 2Ž . i ic where G is the debonding toughness obtained ex- ic perimentally and G is normalized as: i 2 G E 2 t l i m G˜i 23 ' sŽ . b qb Rfs 2 ½ ž / s ž / Rf T lt 1 a a 1 2z m ´ E y yq Rfs 2 f 2 2s 2 T 1yfa a E 1 2 mz ´ q q , 2-1 Ž . ž / 2 f 2s 5 where a’s and b’s are again material- and geometry-relevant parameters given in Hutchinson and Jensen 1990 . Within the two roots of Eq. 2 , Ž . Ž. only the small one is physically meaningful and should be taken as the critical debond length when