3240 HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING matrix at the surface, th(=wrWm at z =I+ h), is free region, ut(=wrWm at 2=1), is h-t[a Er +(b--aEml 2ht[a-Er +(b--cEm] exp(Bi/a)+expAt/a) exp(Bt/a)-exp(-Bt/a (15) 3. 2. Stresses and displacements in the defect-free region(0≤x≤1) When I approaches infinity, u, becomes In this region, the interface remains bonded For a bonded interface, the stress transfer problem has 2ht[a Er +(b--aEml been analyzed and the differential equation govern- B(62-a )Er Em (16) ing the stress distribution in the fiber is [20] d-c (b-=a)Em 3.3. The energy balance condition a er Based on the energy criterion, energy terms are involved: (D)w, the work done by (62(1+n the load,(2) the elastic strain energy. The crack propagation criterion can be when the interface in the effective defect The solution of dr from equation(9)is subjected to subjected to an applied shear stress, t, and the the following two boundary conditions ective defect length h. advances a distance dh. 3.3.1. The work. In the defect region, the work af= Ofd (at2=D) (10a) done, h, due to the applied shear stress, t, at the 10b) interface is Whereas equation (10a) dictates the fiber stress at Wn=2a t(wr -Wm)dz (17) the defect front position, equation (10b)states that the axial stress is zero remote from shear loading. where wr and wm are given by equations(7a-b), re- Using the above boundary conditions, the solution spectively. Substitution of equations (5a-b)and of ar is (7a-b) into equation(17) yields ≤2≤1) 4h32a2+(b2-a2) where B is a dimensionless parameter given bp (11) The change in the work, dWi, in the defect region is hence a-Er+(b--aEr b2(1+m)E[(b2/(b2-a2)ln(b/a)-(3b2-a2)/4b2) The corresponding axial displacement of the fiber relative to the point at z =0 dWh=4nh Er +(b--a2)Em a )Er Em wr=BEr exp(Ba)/a(0s:s). For a constant t, equation(19) is equivalent to the (13) result using dWh=2rahtdun In the defect-free region, the change in the work Similarly, the average axial displacement dn in dw 2zahtdu aomd exp(Bz/a)+exp(-Bz/a (20) BEm exp(Bt/a)-exp(Br/a) ≤z≤D) Substitution of equation(16) into equation(20) Due to the shear load elative displac 4raht-a-Er +(b--a)em between the fiber and the matrix within the defect- dwI=b(b2-aErEm (21)matrix at the surface, uh (=wfÿwm at z = t + h), is uh ˆ h2t‰a2Ef ‡ …b2 ÿ a2†EmŠ a…b2 ÿ a2†EfEm : …8† 3.2. Stresses and displacements in the defect-free region (0RzRt) In this region, the interface remains bonded. For a bonded interface, the stress transfer problem has been analyzed and the di€erential equation govern￾ing the stress distribution in the ®ber is [20] d2 sf dz2 ˆ 1 ‡ …b2 ÿ a2†Em a2Ef  sf    b2…1 ‡ m†  b2 b2 ÿ a2 ln b a  ÿ 3b2 ÿ a2 4b2 : …9† The solution of sf from equation (9) is subjected to the following two boundary conditions: sf ˆ sfd …at z ˆ t† …10a† sf ˆ 0 …at z ˆ 0†: …10b† Whereas equation (10a) dictates the ®ber stress at the defect front position, equation (10b) states that the axial stress is zero remote from shear loading. Using the above boundary conditions, the solution of sf is sf ˆ  exp…bz=a† ÿ exp…ÿbz=a† exp…bt=a† ÿ exp…ÿbt=a†  sfd …0RzRt† …11† where b is a dimensionless parameter given by b ˆ a2Ef ‡ …b2 ÿ a2†Em b2…1 ‡ m†Ef‰…b2=…b2 ÿ a2††ln…b=a† ÿ ……3b2 ÿ a2†=4b2†Š  1=2 : …12† The corresponding axial displacement of the ®ber relative to the point at z = 0 is wf ˆ asfd bEf exp…bz=a† ‡ exp…ÿbz=a† ÿ 2 exp…bt=a† ÿ exp…ÿbt=a† …0RzRt†: …13† Similarly, the average axial displacement of the matrix can be derived, such that wm ˆ asmd bEm exp…bz=a† ‡ exp…ÿbz=a† ÿ 2 exp…bt=a† ÿ exp…ÿbt=a† …0RzRt†: …14† Due to the shear load, the relative displacement between the ®ber and the matrix within the defect￾free region, ut (=wfÿwm at z = t), is ut ˆ 2ht‰a2Ef ‡ …b2 ÿ a2†EmŠ b…b2 ÿ a2†EfEm exp…bt=a† ‡ exp…ÿbt=a† ÿ 2 exp…bt=a† ÿ exp…ÿbt=a† : …15† When t approaches in®nity, ut becomes ut ˆ 2ht‰a2Ef ‡ …b2 ÿ a2†EmŠ b…b2 ÿ a2†EfEm : …16† 3.3. The energy balance condition Based on the energy criterion, the following energy terms are involved: (1) W, the work done by the load, (2) Ue, the elastic strain energy in the ®ber and the matrix, (3) Gi, the interface debond energy. The crack propagation criterion can be established by using the energy balance condition when the interface in the e€ective defect region is subjected to an applied shear stress, t, and the e€ective defect length, h, advances a distance dh. 3.3.1. The work. In the defect region, the work done, Wh, due to the applied shear stress, t, at the interface is Wh ˆ 2pa …t‡h t t…wf ÿ wm† dz …17† where wf and wm are given by equations (7a±b), re￾spectively. Substitution of equations (5a±b) and (7a±b) into equation (17) yields Wh ˆ 4ph3t2 3 a2Ef ‡ …b2 ÿ a2†Em …b2 ÿ a2†EfEm : …18† The change in the work, dWh, in the defect region is hence dWh ˆ 4ph2 t2 a2Ef ‡ …b2 ÿ a2†Em …b2 ÿ a2†EfEm dh: …19† For a constant t, equation (19) is equivalent to the result using dWh=2pahtduh. In the defect-free region, the change in the work, dWt, is dWt ˆ 2pahtdut: …20† Substitution of equation (16) into equation (20) yields dWt ˆ 4paht2 b a2Ef ‡ …b2 ÿ a2†Em …b2 ÿ a2†EfEm dh: …21† 3240 HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING