Pergamon 0956-7151(95)00369X Printed in Great Britain, A 1359-6454961500+0.00 CRACK-WAKE INTERFACIAL DEBONDING CRITERIA FOR FIBER-REINFORCED CERAMIC COMPOSITES CHUN-HWAY HSUEH letals and Ceramics Division, Oak Ridge National Laboratory, PO Box 2008, Oak Ridge, TN37831-6068,USA. (Received 6 July 1995: in revised form 19 September 1995) displacement, and the displacement of the composite due to interfacial debonding. The results are iden to the previous results obtained from a simple approach, in which interfacial debonding is assumed to occur when the mismatch in the axial strain between the fiber and the matrix reaches a critical value Furthermore, the mismatch-strain criterion is found to bear the same physical meaning as the strength-based criterion An important toughening mechanism in fiber- criterion, the analyses using the mismatch. By-based Compared to the analyses using the en strain cr- reinforced ceramic composites is bridging of matrix terion are much less complex [ll]. a question is raised cracks by fibers, which debond from and as to whether the solutions obtained from the mis- frictionally against the matrix [1, 2]. To analyze match-strain criterion agree with those obtained from the toughening effect, a criterion for progres- the energy-based criterion sive debonding at the fiber-matrix interface ac- The strength-based criterion has also been used companied by friction along the debonded interface extensively to study interfacial debonding [12-14 quired However, the solution is always limited to the condition of initial debonding (or frictionless on the fiber to initiate debonding (or the debond debonding). The effect of constant friction along the lyzed (3-9. Depending upon the simplifications analyzed recently [5, and the solution for the adopted in the analysis, different solutions for a, have debond length was identical to that obtained from the been derived. The classical solution [3, 4] was ob lismatch-strain criterion [lI]. A second question is tained by (1)assuming an infinitely long fiber hence raised. Is there a relation between the mis- embedded in a semi-infinite matrix;(2)ignoring the match-strain criterion and the strength-based cri- terion displacement of the fiber portion remaining bonded The purpose of the present study is to address the to the matrix [3] or the strain energy in the matrix [4]; issues mentioned above. First. the stress and the and (3)ignoring Pc s effect. Whereas the appli cation of the classical solution is limited by its displacement distributions in the composite are ana oversimplification, the analysis by Charalambides lyzed, and they are fully determined up to one parameter, the debond length. Whereas the formulas nd Evans [6 is the sensible and simplest one, in e valid for be discussed in the hich only Poisson's cffect is ignored. Thcir analysis following sections, determination of the debond was subsequently modified by Nair [7] to include a length is contingent upon the adoption of constant frictional stress at the debonded interface; debonding criterion. Second, the energy-based cr however, further refinements of the analysis are re- terion is used to analyze the debond length for progressive debonding with a constant friction along An alter native debonding criterion was proposed the debonded interface. Third, the solutions obtained recently [1O]. It was assumed that interfacial debond- from the energy-based criterion are compared to ing occurs when the mismatch in the axial strain those obtained from the mismatch-strain criterion. It between the fiber and the matrix reaches a critical is found that both criteria yield the same solutions value [10]. Based on this assumption, solutions for the Finally, the physical meaning of the approach using fiber-bridging problem have been obtained [11]. the strength-based criterion is discussed, and it is 211
Pergamon Acta mater. Vol. 44, No. 6, pp. 221 l-2216, 1996 Elsevier Science Ltd 0956-7151(95)00369-X Copyright e 1996 Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 1359-6454196 $15.00 + 0.00 CRACK-WAKE INTERFACIAL DEBONDING CRITERIA FOR FIBER-REINFORCED CERAMIC COMPOSITES CHUN-HWAY HSUEH Metals and Ceramics Division, Oak Ridge National Laboratory, PO Box 2008, Oak Ridge, TN 37831-6068, U.S.A. (Received 6 July 1995; in revised form 19 September 1995) Abstract-The condition for progressive debonding with friction along the debonded interface is considered for the bridging fiber in the crack-wake of fiber-reinforced ceramic composites. The energy-based criterion is adopted in the present study to analyze the debond length, the crack-opening displacement, and the displacement of the composite due to interfacial debonding. The results are identical to the previous results obtained from a simple approach, in which interfacial debonding is assumed to occur when the mismatch in the axial strain between the fiber and the matrix reaches a critical value. Furthermore, the mismatch-strain criterion is found to bear the same physical meaning as the strength-based criterion. 1. INTRODUCTION An important toughening mechanism in fiberreinforced ceramic composites is bridging of matrix cracks by fibers, which debond from and slip frictionally against the matrix [l, 21. To analyze the toughening effect, a criterion for progressive debonding at the fiber-matrix interface accompanied by friction along the debonded interface is required. Using the energy-based criterion, the loading stress on the fiber to initiate debonding (or the debond stress for a frictionless interface), cd, has been analyzed [3-91. Depending upon the simplifications adopted in the analysis, different solutions for o,, have been derived. The classical solution [3,4] was obtained by (1) assuming an infinitely long fiber embedded in a semi-infinite matrix; (2) ignoring the displacement of the fiber portion remaining bonded to the matrix [3] or the strain energy in the matrix [4]; and (3) ignoring Poisson’s effect. Whereas the application of the classical solution is limited by its oversimplification, the analysis by Charalambides and Evans [6] is the sensible and simplest one, in which only Poisson’s effect is ignored. Their analysis was subsequently modified by Nair [7] to include a constant frictional stress at the debonded interface; however, further refinements of the analysis are required. An alternative debonding criterion was proposed recently [lo]. It was assumed that interfacial debonding occurs when the mismatch in the axial strain between the fiber and the matrix reaches a critical value [lo]. Based on this assumption, solutions for the fiber-bridging problem have been obtained [ll]. Compared to the analyses using the energy-based criterion, the analyses using the mismatch-strain criterion are much less complex [I 11. A question is raised as to whether the solutions obtained from the mismatch-strain criterion agree with those obtained from the energy-based criterion. The strength-based criterion has also been used extensively to study interfacial debonding [ 12-141. However, the solution is always limited to the condition of initial debonding (or frictionless debonding). The effect of constant friction along the debonded interface on progressive debonding was analyzed recently [15], and the solution for the debond length was identical to that obtained from the mismatch-strain criterion [l I]. A second question is hence raised. Is there a relation between the mismatch-strain criterion and the strength-based criterion? The purpose of the present study is to address the issues mentioned above. First, the stress and the displacement distributions in the composite are analyzed, and they are fully determined up to one parameter, the debond length. Whereas the formulas are valid for all criteria to be discussed in the following sections, determination of the debond length is contingent upon the adoption of the debonding criterion. Second, the energy-based criterion is used to analyze the debond length for progressive debonding with a constant friction along the debonded interface. Third, the solutions obtained from the energy-based criterion are compared to those obtained from the mismatch-strain criterion. It is found that both criteria yield the same solutions. Finally, the physical meaning of the approach using the strength-based criterion is discussed, and it is 2211
22l2 HSUEH: FIBER-REINFORCED CERAMIC COMPOSITES found to be the same as that using the mismatch- 2. 1. Stresses strain criteron When the interface is bonded, the equilibrium axial stresses in the fiber and the matrix, r and om, satisfy 2. THE STRESS AND DISPLACEMENT both the equilibrium and the continuity conditions DISTRIBUTIONS The idealized configuration used in the present (1 analysis is illustrated in Fig. I, where a unidirectional composite is subjected to a tensile load in the direc- tion parallel to the fiber axis, a matrix crack perpe dicular to the loading direction is bridged by intact fibers,which exert a bridging stress, Go, to oppose where Vm Vr)is the volume crack opening. This problem can be modeled by using matrix, and Er and Em are Young's mo a representative volume element shown in Fig. 2. A and matrix, respectively. Combin 如订 the fiber uations fiber with a radius. a is located at the center of a (1)and(2) yields coaxial cylindrical shell of matrix with an outer radius, b, such that a2/b2 corresponds to the volume V Eroo (for bonded interface in the composite [Fig. 2(a) When the interface remains bonded, the composite is subjected to a tensile stress, Rao, and has a displace (for bonded interface) (3b) nent, bonded, in the axial direction [Fig. 2(b)]. In the presence of interfacial debonding, the bridging fiber where F=V,,+vu En. is subjected to a tensile stress, o, and the matrix For a frictional interface, both ar and m vary stress-free at the crack surface [Fig. 2(c)]. Interfacial slowly over distances comparable to the fiber radius bonding and sliding occur along a length, h, with In this case, the characteristics of stresses in any a frictional stress, t, and the end of the debond section transverse to the axial direction can be ap- zone and the crack surface are located at 2 =0 and proximate by a Lame problem, ar and m are h, respectively. The half crack-opening displacement, approximated to be independent of the radial coordi uo, is defined by the relative displacement between the nate [5, 8), and equation (1)is satisfied. The axial fiber and the matrix at the crack surface. Also, stresses in the fiber and the matrix at the end of the ompared to the composite without interfacial debond length, or and omd [Fig. 2( c), can be obtained debonding [Fig. 2(b)], the composite with interfacial from the stresses transfer equation, such that debonding has an additional displacement, debond, in the loading direction [Fig. 2(c) Solutions of at and omd are contingent upon the determination of h. This can be achieved by using the debonding criterion, and the determination of h using the energy-based criterion will be shown in Section 3. With a constant frictional stress, t, the axial stress distributions in the fiber and the matrix, o and a along the debond length are (0≤z≤h)(5a) (0≤z≤h) 2.2. Displacements In the debonded region, the axial displacements in the fiber and the matrix, wr and wm, resulting from the axial stresses described by equations(5a)and(5b)are Fig. 1. A schematic showing a matrix crack bridged by (0≤x≤h)(6a)
2212 HSUEH: FIBER-REINFORCED CERAMIC COMPOSITES found to be the same as that using the mismatchstrain criterion. 2. THE STRESS AND DISPLACEMENT DISTRIBUTIONS The idealized configuration used in the present analysis is illustrated in Fig. 1, where a unidirectional composite is subjected to a tensile load in the direction parallel to the fiber axis. A matrix crack perpendicular to the loading direction is bridged by intact fibers, which exert a bridging stress, e,,, to oppose crack opening. This problem can be modeled by using a representative volume element shown in Fig. 2. A fiber with a radius, a, is located at the center of a coaxial cylindrical shell of matrix with an outer radius, b, such that a2/b2 corresponds to the volume fraction of fibers, Vr, in the composite [Fig. 2(a)]. When the interface remains bonded, the composite is subjected to a tensile stress, Vrcr,, and has a displacement, %ondrd? in the axial direction [Fig. 2(b)]. In the presence of interfacial debonding, the bridging fiber is subjected to a tensile stress, oO, and the matrix is stress-free at the crack surface [Fig. 2(c)]. Interfacial debonding and sliding occur along a length, h, with a frictional stress, z, and the end of the debonding zone and the crack surface are located at z = 0 and h, respectively. The half crack-opening displacement, uO, is defined by the relative displacement between the fiber and the matrix at the crack surface. Also, compared to the composite without interfacial debonding [Fig. 2(b)], the composite with interfacial debonding has an additional displacement, tidebond, in the loading direction [Fig. 2(c)]. t t t Fig. 1. A schematic showing a matrix crack bridged by aligned fibers. 2.1. Stresses When the interface is bonded, the equilibrium axial stresses in the fiber and the matrix, uf and G,, satisfy both the equilibrium and the continuity conditions, such that v,cr,+ v,cr, = V@” (1) or elll E,=E, (2) where V,,, (= 1 - V,) is the volume fraction of the matrix, and Ef and Em are Young’s moduli of the fiber and matrix, respectively. Combination of equations (1) and (2) yields VrEre, er=7- (for bonded interface) (3a) Vr-K 00 em = E, (for bonded interface) (3b) where EC = Vf Ei + V,,, Em. For a frictional interface, both err and em vary slowly over distances comparable to the fiber radius. In this case, the characteristics of stresses in any section transverse to the axial direction can be approximated by a Lame problem, of and em are approximated to be independent of the radial coordinate [5, 81, and equation (1) is satisfied. The axial stresses in the fiber and the matrix at the end of the debond length, ofd and umd [Fig. 2(c)], can be obtained from the stresses transfer equation, such that 2hr Ofd = O” - 7 (44 2h Vf7 fJmd=-. aVm (4b) Solutions of crrd and emd are contingent upon the determination of h. This can be achieved by using the debonding criterion, and the determination of h using the energy-based criterion will be shown in Section 3. With a constant frictional stress, 7, the axial stress distributions in the fiber and the matrix, uf and em, along the debond length are or = bid + z(go;ofd) (O 1 -f rJrnd (O<z <h). (5b) 2.2. Displacements In the debonded region, the axial displacements in the fiber and the matrix, wr and w,, resulting from the axial stresses described by equations (5a) and (5b) are w,=zard+ Ef z'("O-ufd) co<_ <hl 2hE, ” ’ (64
HSUEH: FIBER-REINFORCED CERAMIC COMPOSITES 2213 bonded vOo debond (a) Prior to loading (b) Loading without debonding (c) Loading with debonding tive volume element for the fiber bridging problem: (a)prior to loading;(b) loading without interfacial debonding; and (c) loading with interfacial debonding. The half cracking opening displacement, uo, and the displacement of the composite due to interfacial debonding. udehond, are also applied stress, ao, and the debond length, h, advances (0≤z≤h)(6b) a distance dh The half crack opening displacement, uo(=w -Wm at 3. 1. The elastic strain energy z=h), becomes [Fig. 2(c) In the bonded region, the elastic strain energy ho h'tE density in the fiber and the matrix is of/2E and (7)0m/2Em, respectively, where or and Om are given by er am ee equations(3a)and(3b). When the debonded length dvances a distance dh, the bonded region is de- displacement in the composite, We, within a length h the fiber and the matrix have the volume fractions ve IFig. 2(b) and Vm, respectively, the change in the elastic strait hv energy, dErby, in the bonded region due to the Wc()=e (8)advance of the debond length is the additional axial displacement of the com- dUeh= due to debonding, Debond [=w(h)-w(h)], 2(+)0 [Fg.2(c) Substitution of equations(3a)and(3b)il (10) yields hVm End ht Debond ERe rta'vro, dh Solutions of Mo and waebnd are also contingent upon In the bonded region, the elastic strain energy in the determination of the debond length, h he fiber and the matrix, Uedi, is Ued, ib-lrrafvmo2 3. THE ENERGY-BASED CRITERION Er E eneray derm s ha e nevgoyve: d oriente chastic swan respectively. Substituting equations (4)and (5) w, the work done by the applied stress. The debond 64(3m-5+04+ ing criterion can be established by using the energy balance condition when the fiber is subjected to an +a(3
HSUEH: FIBER-REINFORCED CERAMIC COMPOSITES 2213 (a) Prior to loading (b) Loading without debonding (c) Loading with debonding Fig. 2. A representative volume element for the fiber-bridging problem: (a) prior to loading; (b) loading without interfacial debonding; and (c) loading with interfacial debonding. The half cracking opening displacement, u,,, and the displacement of the composite due to interfacial debonding, udebond, are also shown. (6b) The half crack opening displacement, u,, (= wt - NJ,,, at z = h), becomes [Fig. 2(c)] (7) In the absence of interfacial debonding, the axial displacement in the composite, w,, within a length h is [Fig. 2(b)] ii Vroo w,(h) = E. C (8) Hence, the additional axial displacement of the composite due to debonding, udebond [ = w,(h) - w,(h)], becomes [Fig. 2(c)] Solutions of u0 and udeband are also contingent upon the determination of the debond length, h. 3. THE ENERGY-BASED CRITERION Based on the energy-based criterion, the following energy terms are involved: (1) U,, the elastic strain energy in the fiber and the matrix; (2) U,, the energy due to sliding at the debonded interface; (3) Gi, the energy release rate for interfacial debonding; and (4) W, the work done by the applied stress. The debonding criterion can be established by using the energy balance condition when the fiber is subjected to an applied stress, g,,, and the debond length, h, advances a distance dh. 3. I. The elastic strain energy In the bonded region, the elastic strain energy density in the fiber and the matrix is az/2E, and $,J2E,,,, respectively, where cf and cr,,, are given by equations (3a) and (3b). When the debonded length advances a distance dh, the bonded region is decreased by a length dh and by a volume zb2 dh. Since the fiber and the matrix have the volume fractions Vf and V,, respectively, the change in the elastic strain energy, dU+, , in the bonded region due to the advance of the debond length is d,,,=+(~+~)dh. (10) Substitution of equations (3a) and (3b) into equation (10) yields d Ue(b, = -na2Vfai dh 2E, ’ (11) In the bonded region, the elastic strain energy in the fiber and the matrix, Uetdj, is where of and Q, are given by equations (5a) and (5b), respectively. Substituting equations (4) and (5) into equation (12) UeCdj becomes 4h3T2V + I azv,E,’ 1 (13)
HSUEH: FIBER-REINFORCED CERAMIC COMPOSITES When the debond length advances a distance dh, the 3. 4. The interfacial energy change in the elastic strain energy, duddy in the When the debond length advances a distance di debonded region can be obtained by taking the the surface area of the debonded interface is increased derivative of equation(13)with respect to h, and the by 2Ta dh. The change in the interfacial energy, dG result is dG=2rag dh due -40++nE The debonding criterion can be obtained from the The change in the elastic strain energy in the energy balance condition, such that composite dU(=dUba+dUed ) is hence dW=dU。+dU、+dG; (24) due -man Em oo-2he dh.(15)Substitution of equations(15).(18),(21)and(23) into 2EE Eecg 2/TE 3. 2. The sliding energy VE During interfacial sliding, energy is dissipated due Equation(25)defines the relation between the bridg to the relative displacement between the fiber and the ing stress, oo, the interface energy, Gi. and the under an interfacial frictional stress, t. The (frictional) debond length, h sliding energy at the debonded interface, U,,is The stress required for initial debonding(or fric Us=rat(or-Wm)d (16)equation (25)by letting h=0(or r =0), such that EEg here wr and wm are given by equations(6a)and(6b), respectively. Substitution of equations(4)and (6)into Equation (26) is identical to the result derived by equation(16) yields Charalambides and Evans [6]. Combining equations (25)and(26), the debonded length, h, becomes U hoa hte 2e avM E Em (17) avm Em(oo-ga) I he change in the sliding energy, dUs. is hence Substitution of equations(26)and(27)into equations d=2(m-21)u.(1sy) avm E o For a constant t, equation(18)is equivalent to the esult using dUs= 2raht duay ak2 E2 EG ErEC 3.3. The work In the absence of interfacial bonding (i.e With the bridging stress, ao, on the fiber, the work equations(28a)and(28b)become done due to interfacial debonding, w, is eReCt Vm a The change in the work done is hence dw=rago dudebe (20) Equations(29a)and(29b) are identical to the dis- (20)placements derived in the MCE [16] and the ACK Substitution of equation (9)into equation (20) yields [17 models, respectively. Whereas waebond is cor sidered in the ACK model [ 17], uo is considered in the dw=ta'a v lin o 2hr\ MCE model [16]. However, this difference has not Ee can be seen from equations(15),(18)and(21) 4. COMPARISON WITH THE MISMATCH-STRAIN CRITERION dw-dU ng criterion has been proposed dU= ial debonding occurs when the mismatch in the axial strain between the fiber and the The above relation is valid for an elastic system matrix reaches a critical value [10]. Based on this
2214 HSUEH: FIBER-REINFORCED CERAMIC COMPOSITES When the debond length advances a distance dh, the change in the elastic strain energy, dU+,,, in the debonded region can be obtained by taking the derivative of equation (13) with respect to h, and the result is s dh. (14) m m 1 The change in the elastic strain energy in the composite, dU,( =dU,,,, + dU,,,,), is hence dUc=s(o,-sydh. (15) 3.2. The sliding energy During interfacial sliding, energy is dissipated due to the relative displacement between the fiber and the matrix under an interfacial frictional stress, z. The sliding energy at the debonded interface, U,, is U, = 2na s ,r z(wf- w,)dz, (16) 0 where wt and w, are given by equations (6a) and (6b), respectively. Substitution of equations (4) and (6) into equation (16) yields Usi,= 2nar !+L 2h %E, 2E, > 3aV,,,E,E,,, (17) The change in the sliding energy, dU,, is hence dU, = 2zaT 2 - f 3)dh. (18) For a constant r, equation (18) is equivalent to the result using dU, = 2nahr du,. 3.3. The work With the bridging stress, go, on the fiber, the work done due to interfacial debonding, W, is W = na *oO udebond (1% The change in the work done is hence d W = xa *oO dudebond. Substitution of equation (9) into equation (20) yields dh. (21) It can be seen from equations (15), (18) and (21) that dU,= dW-dU, 2 (22) The above relation is valid for an elastic system 3.4. The interfacial energ? When the debond length advances a distance dh, the surface area of the debonded interface is increased by 2na dh. The change in the interfacial energy, dG,, is hence dG, = 2zaG, dh. (23) 3.5. The energy balance condition and solutions The debonding criterion can be obtained from the energy balance condition, such that dW=dU,+dU,+dG,. (24) Substitution of equations (15) (18) (21) and (23) into equation (24) yields E,E,G, I,’ a,=2 ~ i > + 2hzE, aV,-% aV,E,,,’ (25) Equation (25) defines the relation between the bridging stress, do, the interface energy, G,, and the (frictional) debond length, h. The stress required for initial debonding (or frictionless debonding), rr,, can be obtained from equation (25) by letting h = 0 (or 7 = 0), such that (26) Equation (26) is identical to the result derived by Charalambides and Evans [6]. Combining equations (25) and (26) the debonded length, h, becomes Substitution of equations (26) and (27) into equations (7) and (9) yields (28a) aV~E~a~ Vm 6, G, ‘debond = 4E,E; f (28b) In the absence of interfacial bonding (i.e. G, = 0). equations (28a) and (28b) become Udebond = 4E, Ef 7 Equations (29a) and (29b) are identical to the displacements derived in the MCE [16] and the ACK [17] models, respectively. Whereas udebond is considered in the ACK model [ 171, u0 is considered in the MCE model [16]. However, this difference has not been recognized by many researchers. 4. COMPARISON WITH THE MISMATCH-STRAIN CRITERION A simple debonding criterion has been proposed such that interfacial debonding occurs when the mismatch in the axial strain between the fiber and the matrix reaches a critical value [IO]. Based on this
HSUEH: FIBER-REINFORCED CERAMIC COMPOSITES 2215 a(0-0d) with friction along the debonded interface has been (33) derived[11] which is reviewed and compared with the Substitution of equation(32a)into equation(33) present analysis as follows When the bridging stress in the fiber reaches th VmEm(o-ad) debond stress, dd, interfacial debonding initiates at h (34) 2ET the crack surface(where the matrix is stress-free) and Equation(34)is identical to the results oblained fruil the critical mismatch strain, (d, is both the energy-based [see equation(27)] and the strength-based criteria [15]. Both uo an (30) also been derived using the mismatch-strain criterion During subsequent loading (i.e, o >(d), interfacial [II], and they are identical to those obtained in the debonding extends underneath the crack surface. Due to the stress transfer from the fiber to the matrix 5. THE STRENGTH-BASED CRITERION through the intertacial shear stress, stresses exist in For the strength-based criterion, the shear lag the matrix underneath the crack surface. The axial model[18] has been used extensively to analyze the stresses in the fiber and the matrix at the end of the interfacial shear stresses when the fiber is subjected to debonding zone are assumed to be ord and omd, axial loading, and interfacial debonding occurs when respectively, and they satisfy the mechanical equi- the interfacial shear strength. T,. Is reached, Inter- librium condition depicted by equation (1).The mis- facial debonding initiates at the loaded surface when match-strain criterion requires that the mismatch in the loading stress on the fiber reaches oa,at which the the axial strain between the fiber and the matrix at the end of the debonding zone reaches ed: i.e corresponding maximum interfacial shear stress reaches t a difference has been noted betwee 3D)underneath the loaded surface [19]. Whereas the matrix is stress-free at the loaded surface Combination of equations (I),(30) and (3m) yields jected to axial stresses underneath the loaded surface V Ergo+Vema due to the stress transfer from the fiber to the matrix (32a) Hence, the magnitude of the interfacial shear stress V Em(oo-0a) induced by a loading stress oa on the fiber at (32b) loaded surface is different from that induced by an axial stress od in the fiber underneath the loaded With a constant frictional stress, t, along the debond surface. Assuming that the axial stresses at the end of from oo at the crack surface to ord at the end of the the fiber and the matrix, the relation between rd and debonding zone. The debond length, h, is related to a can be derived using the strength-based criterion the bridging stress, oo, and rd by and this is shown as follows ofd-Efond fd Em om omd Fig 3. The procedures in deriving the interfacial shear stress at the end of the debonding zone: (a) tractions of Eromd/Em and omd are imposed on the fiber and the matrix, respectively, resulting in a uniform axi stress is induced;(c)combination of the aboy Epmd/Em is imposed on the fiber, and the interfacial shear train in the composite(b)a traction of a e two procedures results in the condition of tractions at the nd of the debonding zone
HSUEH: FIBER-REINFORCED CERAMIC COMPOSITES 2215 criterion, the condition for progressive debonding with friction along the debonded interface has been derived [ 1 l] which is reviewed and compared with the present analysis as follows. When the bridging stress in the fiber reaches the debond stress, gd, interfacial debonding initiates at the crack surface (where the matrix is stress-free), and the critical mismatch strain, cd, is EdA. E f During subsequent loading (i.e. rrO > crd), interfacial debonding extends underneath the crack surface. Due to the stress transfer from the fiber to the matrix through the interfacial shear stress, stresses exist in the matrix underneath the crack surface. The axial stresses in the fiber and the matrix at the end of the debonding zone are assumed to be o, and emd, respectively, and they satisfy the mechanical equilibrium condition depicted by equation (1). The mismatch-strain criterion requires that the mismatch in the axial strain between the fiber and the matrix at the end of the debonding zone reaches td; i.e. Ofd *md cd=--- 4 Em’ (31) Combination of equations (I), (30) and (31) yields I’r Era, + v, E, od Ofd = EC (324 Vf Em (00 - ud ) Ornd = 4 . Wb) With a constant frictional stress, r, along the debond length, h, the axial stress in the fiber decreases linearly from u0 at the crack surface to old at the end of the debonding zone. The debond length, h, is related to the bridging stress, oO, and gfd by + h= a(ail - bfd) 2r (33) Substitution of equation (32a) into equation (33) yields h= ~f'm-Qn(~,- Od) 2E,r (34) Equation (34) is identical to the results obtained from both the energy-based [see equation (27)] and the strength-based criteria [15]. Both u0 and &+sond have also been derived using the mismatch-strain criterion [ll], and they are identical to those obtained in the present study. 5. THE STRENGTH-BASED CRITERION For the strength-based criterion, the shear lag model [18] has been used extensively to analyze the interfacial shear stresses when the fiber is subjected to axial loading, and interfacial debonding occurs when the interfacial shear strength, z,, is reached. Interfacial debonding initiates at the loaded surface when the loading stress on the fiber reaches crd, at which the corresponding maximum interfacial shear stress reaches t,. A difference has been noted between debonding at the loaded surface and debonding underneath the loaded surface [19]. Whereas the matrix is stress-free at the loaded surface, it is subjected to axial stresses underneath the loaded surface due to the stress transfer from the fiber to the matrix. Hence, the magnitude of the interfacial shear stress induced by a loading stress rrd on the fiber at the loaded surface is different from that induced by an axial stress ~~ in the fiber underneath the loaded surface. Assuming that the axial stresses at the end of the debonding zone are ofd and rrmdr respectively, in the fiber and the matrix, the relation between gfd and od can be derived using the strength-based criterion and this is shown as follows. afd _ Ef%d t Em Ofd Omd t arnd Fig. 3. The procedures in deriving the interfacial shear stress at the end of the debonding zone: (a) tractions of Ep,,/E, and emd are imposed on the fiber and the matrix, respectively, resulting in a uniform axial strain in the composite; (b) a traction of e fd - Ef~md/Em is imposed on the fiber, and the interfacial shear stress is induced; (c) combination of the above two procedures results in the condition of tractions at the end of the debonding zone
HSUEH: FIBER-REINFORCED CERAMIC COMPOSITES At the end of the debonding zone the interfacial debonded interface. However, this nonuniform fric shear stress can be analyzed using the following tional stress can be simplified by using an averaging procedures. First, tractions of Eomd /Em and md are technique, in which the average frictional stress along imposed on the fiber and the matrix, respectively the sliding interface is considered in the analysis [11] [Fig 3(a)). This would result in a uniform axial strain Poisson's effect on the relation between the bridging ame/Em in the composite, and no interfacial shear stress and the debond length has been addressed stress is induced. Then, a traction of j-eromd/ Em elsewhere [11]. It is also noted that residual stresses he fiber, and this would induce the are not included in the present analysis. Effects of above two procedures, the tractions imposed on the addressed elsewhere[6,8, 2, y Bonding have been interfacial shear stress [Fig. 3(b). Combining the residual stresses on interfacial det fiber and the matrix are agd and Omd, respectively crack surface if a traction of ga-eromd/Em is im- thanks Dr E. Lara-Curzio posed on the fiber alone. To satisfy the debonding by the U.s. Department of Energy, Division following relation is hence required Er (35) Ceramic Composites Program, under contract DE-ACO5- 840R21400 with Lockheed marti It is noted that equation ( 35)can also be obtained by combining equations(30) and (31), which are derived using the mismatch-strain criterion. Hence, 1. A G. Evans and R. M. McMeeking, Acta metall. 34, the strength-based criterion yields the same result [i.e. 2435(1986) ation (34) as that using the mismatch-strain 2. P F. Becher, C. H. Hsueh, P nd T N. Tiegs criterion. This is not surprising. Since the interfacial 3. C. Gurney and 3. lunt, Proc.R.Soc.Lond.A299,508 shear stress results from the tendency of a relative (196 displacement in the axial direction between the fiber 4. J. P. Outwater and M. C. Murphy, in Proc. 24th Annual Technical Conf Reinforced Plastic/Composites Division alent to the mismatch-strain criterion. It is also noted The Society of the Plastics Industrial Inc, Composites that, whereas d is related to the energy release rat Division, New York, Paper No. llc(1969) S.Y. C. Gao. Y. w. Mai and B, Cotterell, J, for interfacial debonding, G;, in the energy-based Phys.(ZAMP)39, 550(1988) riterion, it is related to the interfacial shear strength, 6. P. G Charalambides and A. G. Evans, J. Am. Ceran gth-based [2-14] 7. S. V. Nair, J. Am. Ceram. Soc. 73, 2839(1990) 8.J utchinson and H. M. Jensen, Mech. Mater. 9 CONCLUDING REMARKS 139(1990 9. C. H. Hsueh, Mater. Sci. Engng A159, 65(1992) Using the energy-based criterion, progressive N. Shafry, D G. Brandon and M. Terasaki, Euro-Cer debonding with friction along the debonded interface cs3,3.453(l98 II. C, H. Hsueh, Mater 30,1781(1995) is analyzed for the bridging fiber in the crack-wake of 12. P. Lawrence, J. Mater. Sci.7, 1(197 fiber-reinforced ceramic composites. It is noted that 13. A. Takaku and R. G. C. Arridge, Phys. D: App/ he displacement term involved in calculating the Phys.6,2038(1973 work done by the load is the displacement of the 14. C H Hsueh, Mater. Sci. Engng A123, 1(1990) composite due to interfacial debonding, not the crack 15. B. Budiansky. A G. Evans and J. W. Hutchinson. Solids Struct, 32, 315(1995) opening displacement. The present results for crack- 16. D. B. Marshall, B. N. Cox and A.G.Evans.Acta ake interfacial debonding are identical to those 3,2013(1985) obtained from the mismatch-strain criterion, in which 17. J. Aveston, G A. Cooper and A. Kelly, in Conf. Proc. interfacial debonding is assumed to occur when the mismatch in the axial strain between the fiber and the (1971) IPC Science and Technology Press Ltd matrix reaches a critical value. Also, the mismatch- 18. H. L. Cox, Brit. J. app/. Phys. 3, 72(1952 and G. w. Groves. J. M c.14,43 It is noted that a constant Trictional stress is 21. P D Jero, R.J. Kerans and T.A.Parthasarathy,J.Am assumed in the present analysis. The assumption of a Ceran.Soc.74,2793(1991). constant frictional stress is appropriate when friction 22. T.J. Mackin, P. D. Warren and A. G. Evans, Acla results from the interfacial roughness effect [20-22] 25!(1992) In the case of Coulomb friction with Poisson's effect C. H. Hsueh, Mater. S ngAl45,13(1991) R.J. Kerans and T A arathi, J. Am. Ceram he frictional stres (1991)
2216 HSUEH: FIBER-REINFORCED CERAMIC COMPOSITES At the end of the debonding zone, the interfacial shear stress can be analyzed using the following procedures. First, tractions of E+r,,/E, and omd are imposed on the fiber and the matrix, respectively [Fig. 3(a)]. This would result in a uniform axial strain a,,/E,,, in the composite, and no interfacial shear stress is induced. Then, a traction of old - E,o,,/E, is imposed on the fiber, and this would induce the interfacial shear stress [Fig. 3(b)]. Combining the above two procedures, the tractions imposed on the fiber and the matrix are cfd and crmd, respectively [Fig. 3(c)]. Hence, the interfacial shear stress at the end of the debonding zone is equivalent to that at the crack surface if a traction of ofd - Efa,,/E,,, is imposed on the fiber alone. To satisfy the debonding condition at the end of the debonding zone, the following relation is hence required: (35) It is noted that equation (35) can also be obtained by combining equations (30) and (31), which are derived using the mismatch-strain criterion. Hence, the strength-based criterion yields the same result [i.e. equation (34)] as that using the mismatch-strain criterion. This is not surprising. Since the interfacial shear stress results from the tendency of a relative displacement in the axial direction between the fiber and the matrix, the strength-based criterion is equivalent to the mismatch-strain criterion. It is also noted that, whereas cd is related to the energy release rate for interfacial debonding, G,, in the energy-based criterion, it is related to the interfacial shear strength, z,, in the strength-based criterion [12-141. 6. CONCLUDING REMARKS Using the energy-based criterion, progressive debonding with friction along the debonded interface is analyzed for the bridging fiber in the crack-wake of fiber-reinforced ceramic composites. It is noted that the displacement term involved in calculating the work done by the load is the displacement of the composite due to interfacial debonding, not the crack opening displacement. The present results for crackwake interfacial debonding are identical to those obtained from the mismatch-strain criterion, in which interfacial debonding is assumed to occur when the mismatch in the axial strain between the fiber and the matrix reaches a critical value. Also, the mismatchstrain debonding criterion is found to have the same physical meaning as the strength-based debonding criterion. It is noted that a constant frictional stress is assumed in the present analysis. The assumption of a constant frictional stress is appropriate when friction results from the interfacial roughness effect [20-221. In the case of Coulomb friction with Poisson’s effect, the frictional stress is nonuniform along the debonded interface. However, this nonuniform frictional stress can be simplified by using an averaging technique, in which the average frictional stress along the sliding interface is considered in the analysis [l 11. Poisson’s effect on the relation between the bridging stress and the debond length has been addressed elsewhere [ll]. It is also noted that residual stresses are not included in the present analysis. Effects of residual stresses on interfacial debonding have been addressed elsewhere [6, 8,23.24]. Acknowledgements-The author is indebted to Professor S. V. Nair for bringing the subject to his attention, and thanks Dr E. Lara-Curzio and Dr S. Raghuraman for reviewing the manuscript. Research sponsored jointly by the U.S. Department of Energy, Division of Materials Sciences, Office of Basic Energy Sciences, and Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Industrial Technologies, Industrial Energy Efficiency Division and Continuous Fiber Ceramic Composites Program, under contract DE-ACOS- 840R21400 with Lockheed Martin Energy Systems. 5. 6. I. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. REFERENCES A. G. Evans and R. M. McMeeking, Acta merail. 34, 2435 (1986). P. F. Becher, C. H. Hsueh, P. Angelini and T. N. Tiegs. J. Am. Ceram. Sot. 71, 1050 (1988). C. Gurney and J. Hunt, Proc. R. Sot. Lond. A299, 508 (1967). J. P. Outwater and M. C. Murphy, in Proc. 24th Annual Technical Conf: Reinforced Plastic/Composites Diuision, The Society of the Plastics Industrial Inc., Composites Division, New York, Paper No. llc (1969). Y. C. Gao, Y. W. Mai and B. Cotterell, J. appl. Math. Phys. @AMP) 39, 550 (1988). P. G. Charalambides and A. G. Evans, J. Am. Ceram. Sot. 72, 746 (1989). S. V. Nair, J. Am. Ceram. Sot. 73, 2839 (1990). J. W. Hutchinson and H. M. Jensen, Mech. Mater. 9, 139 (1990). C. H. Hsueh, Mater. Sci. Engng A159, 65 (1992). N. Shafry, D. G. Brandon and M. Terasaki, Euro-Ceramics 3, 3.453 (1989). C. H. Hsueh, J. Mater. Sci. 30, 1781 (1995). P. Lawrence, J. Mafer. Sci. 7, 1 (1972). A. Takaku and R. G. C. Arridge, J. Phys. D: Appl. Phys. 6, 2038 (1973). C. H. Hsueh, Mater. Sci. Engng A123, 1 (1990). B. Budiansky, A. G. Evans and J. W. Hutchinson, Inr. J. Solids Struct. 32, 315 (1995). D. B. Marshall, B. N. Cox and A. G. Evans, Acta metall. 33, 2013 (1985). J. Aveston, G. A. Cooper and A. Kelly, in Conf: Proc. National Physical Laboratory, Guildford, pp. 15-26. IPC Science and Technology Press Ltd., Guildford (1971). H. L. Cox, Brit. J. appl. Phys. 3, 72 (1952). J. K. Kim, C. Baillie and Y. W. Mai, J. Mater. Sri. 27, 3143 (1992). J. Bowling and G. W. Groves, J. Mater. Sci. 14, 431 (1979). P. D. Jero, R. J. Kerans and T. A. Parthasarathy, J. Am. Ceram. Sot. 74, 2193 (1991). T. J. Mackin. P. D. Warren and A. G. Evans. Acta metall. mater.‘40, 1251 (1992). C. H. Hsueh, Mater. Sci. Engng A145, 13 (1991). R. J. Kerans and T. A. Parthasarathy, J. Am. Gram. Sot. 74, 1585 (1991)