- Email: [email protected]
Synergistic effects of fiber debonding and fracture on matrix cracking in fiber-reinforced ceramic-matrix composites
Materials Science & Engineering A 682 (2017) 482–490
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Synergistic effects of fiber debonding and fracture on matrix cracking in fiber-reinforced ceramic-matrix composites
crossmark
Li Longbiao College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, No. 29 Yudao St., Nanjing 210016, PR China
A R T I C L E I N F O
A BS T RAC T
Keywords: Ceramic-matrix composites (CMCs) Matrix cracking Interface debonding Fiber failure
In this paper, the synergistic effects of fiber debonding and fracture on matrix cracking stress of fiber-reinforced ceramic-matrix composites (CMCs) have been investigated using the energy balance approach. The shear-lag model cooperated with fiber/matrix interface debonding criterion and fiber fracture model has been adopted to analyze stress distribution in CMCs. The relationships between matrix cracking stress, interface debonding and slipping, and fiber fracture have been established. The effects of fiber volume fraction, interface shear stress, interface debonded energy, fiber Weibull modulus, and fiber strength on matrix cracking stress, interface debonded length and fiber broken fraction have been analyzed. The experimental matrix cracking stress of three different CMCs, i.e., SiC/borosilicate, SiC/LAS, and C/borosilicate, with different fiber volume fraction have been predicted.
1. Introduction Ceramic materials possess high strength and modulus at elevated temperature. But their use as structural components is severely limited because of their brittleness. Continuous fiber-reinforced ceramicmatrix composites, by incorporating fibers in ceramic matrices, however, not only exploit their attractive high-temperature strength but also reduce the propensity for catastrophic failure. These materials have already been implemented on some aero engines’ components [1]. The CMCs exhibit distinct behaviors at stresses above and below the matrix cracking stress, which is associated with the onset of matrix cracking and with the formation of hysteresis loops that results from matrix cracking and frictional slipping of the fibers bridging matrix cracks. In the environmentally-stable fibers and fiber coating in oxidizing environments, the matrix cracking stress has long been considered the maximum allowable design stress for CMCs in the real applications involving oxidizing environments. Many researchers performed experimental and theoretical investigations on matrix cracking of fiber-reinforced CMCs. For analytical modeling, the energy balance approach developed by Aveston, Cooper and Kelly (ACK) [2], Budiansky, Hutchinson and Evans (BHE) [3], and Chiang [4], and the fracture mechanics approach proposed by Marshall, Cox and Evans (MCE) [5], and McCartney [6] have been used to investigate the matrix cracking stress. The analytical results show that the matrix cracking stress was closely related with the interface friction stress. The composite with the higher interface shear stress results in the higher matrix cracking stress. When the fiber/
matrix interface is weakly bonding, the BHE model will reduce to the ACK model, and when the interface is strongly bonding, the matrix cracking stress predicted by the BHE model is the same with that of Aveston and Kelly [7]. Rajan and Zok [8] investigate the mechanics of a fully bridged steady-state matrix cracking in unidirectional CMCs under shear loading. However, the models mentioned above do not consider the synergistic effects of fiber debonding and fracture on matrix cracking stress in CMCs. During the process of matrix cracking in CMCs, the fiber/matrix interfacial debonding occurs due to the fiber-matrix relative displacement above the matrix cracking plane, and fibers failure also occurs due to the statistical properties of the fibers strength. In the present analysis, the shear-lag model is adopted to analyze the micro-stress field of the damaged composite, including the fiber, matrix and fiber/ matrix interface shear stress in the interface debonded and bonded region. The Global Load Sharing criterion (GLS) and the fracture mechanics approach were used to determine the broken fibers fraction and the interface debonded length during matrix cracking. The effects of interface debonding and slipping, and fibers fracture have been taken into consideration to solve the matrix cracking stress. The influence of material properties, i.e., fiber volume fraction, interface shear stress, interface debonded energy, fiber Weibull modulus, and fiber strength on matrix cracking stress, interface debonded length and fiber broken fraction have been analyzed.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.msea.2016.11.077 Received 20 September 2016; Received in revised form 14 October 2016; Accepted 22 November 2016 Available online 23 November 2016 0921-5093/ © 2016 Published by Elsevier B.V.
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
2. Materials and experimental procedures The three unidirectional fiber-reinforced CMCs for matrix cracking were provided by Barsoum et al. [9], including (a) the SiC/lithium/ aluminosilicate (SiC/LAS) system, which comprises a SiC fiber (Nicalon, Nippon Carbon Co., Tokyo Japan) and a LAS matrix; (b) the carbon/borosilicate system, which is made of HMU carbon fibers (HMU Hercules carbon fiber) embedded in a borosilicate glass matrix; and (c) the SiC/borosilicate system, consisting of a SiC monofilament (SCS-6 SiC Fiber, Textron Specialty Materials, Lowell, Massachusetts, USA) embedded in the same borosilicate glass matrix. The variation in fiber volume fraction is achieved by controlling the amount of glass powder used in the preparation of the polymer slurry. Three-point-bend test specimens were performed in the form of flat bars having the dimensions of 3×0.5×0.18 cm3 for SiC/LAS, 3×0.5×0.2 cm3 for C/borosilicate, and 6×0.5×0.2 cm3 for SiC/borosilicate. The pin-support span of the three-point bend fixture was 2.54 cm for the shorter bars and 5.2 cm for the longer bars. During the three-point-bend testing, the maximum beam deflection was measured with an extensometer mounted at the beam mid-span; at the same time, the electrical resistance of the gold film sputtered on the polished specimen surface was measured with a digital Ohm meter. The deflection and electrical resistance were recorded as a continuous function of loading and were then used to determine the onset of the matrix cracking stress.
Fig. 1. The schematic of crack-tip and interface debonding.
Substituting Eqs. (2) and (3) into the Eq. (1), it leads into the form of
⎡ ⎛ T ⎞m +1⎤ ⎫ ⎛ σ ⎞m +1 ⎧ ⎪ ⎪ σ ⎬ = T ⎜ c⎟ ⎨ 1 − exp ⎢ −⎜ ⎟ ⎥ ⎪ ⎪ ⎝T ⎠ ⎩ Vf ⎣⎢ ⎝ σc ⎠ ⎥⎦ ⎭
Using the Eq. (4), the stress T carried by intact fibers at the matrix cracking plane can be determined. Substituting the intact fiber stress T into the Eq. (2), the relationship between the fiber failure probability and applied stress can be determined.
3. Stress analysis When fibers break during matrix cracking, the loads dropped by the broken fibers must be transferred to the intact fibers in the crosssection. Two dominant failure criterions are present in the literatures for modeling fibers failure, i.e., Global Load Sharing criterion (GLS) and Local Load Sharing criterions (LLS). The GLS criterion assumes that the load from any one fiber is transferred equally to all other intact fibers in the same cross-section plane. The GLS assumption neglects any local stress concentrations in the neighborhood of existing breaks, and is expected to be accurate when the interfacial shear stress is sufficiently low. Models that include GLS explicitly have been developed, which includes Thouless and Evans [10], Cao and Thouless [11], Sutcu [12], Schwietert and Steif [13], Curtin [14], Weitsman and Zhu [15], Hild et al. [16], Solti et al. [17], Cho [18], Paar et al. [19], Liao and Reifsnider [20], and so on. The LLS assumes that the load from the broken fiber is transferred to the neighborhood intact fibers, and is expected to be accurate when the interface shear stress is sufficiently high. Models that include LLS explicitly have been developed, which includes Zhou and Curtin [21], Dutton et al. [22], Xia and Curtin [23], and so on. The two-parameter Weibull model is adopted to describe the fiber strength distribution, and the Global Load Sharing (GLS) assumption is used to determine the load carried by intact and fracture fibers [14].
σ = T [1 − P (T )] + Tb P (T ) Vf
3.1. Downstream stresses The composite with fiber volume fraction Vf is loaded by a remote uniform stress σ normal to a long crack plane, as shown in Fig. 1. The unit cell in the downstream Region I contained a single fiber surrounded by a hollow cylinder of matrix is extracted from the ceramic composite system, as shown in Fig. 2. The fiber radius is rf, and the matrix radius is R (R=rf/Vf1/2). The length of the unit cell is half matrix crack spacing lc/2, and the interface debonded length is ld. In the debonded region, the interface is resisted by τi. For the debonded region in Region I, the force equilibrium equation of the fiber is given by Eq. (5) [3].
dσf (z ) 2τ (z ) =− i dz rf
σf (z = 0) = T
(6)
σm (z = 0) = 0
(7)
The total axial stresses in Region I satisfy the Eq. (8).
(1)
(2)
where m denotes the fiber Weibull modulus; and σc denotes the fiber characteristic strength. The load carried by broken fibers is determined by the Eq. (3).
⎡ ⎛ σ ⎞m +1 1 − P (T ) ⎤ ⎥ Tb = T ⎢ ⎜ c ⎟ − ⎢⎣ ⎝ T ⎠ P (T ) ⎥⎦
(5)
The boundary conditions of the fiber and matrix axial stresses at the crack plane (i.e., z=0) are given by
where Vf denotes the fiber volume fraction; T denotes the load carried by intact fibers; Tb denotes the load carried by broken fibers; and P(T) denotes the fiber failure probability.
⎡ ⎛ T ⎞m +1⎤ P (T ) = 1 − exp ⎢ −⎜ ⎟ ⎥ ⎢⎣ ⎝ σc ⎠ ⎥⎦
(4)
(3)
Fig. 2. The schematic of shear-lag model considering interface debonding.
483
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
Vf σf (z ) + Vm σm (z ) = σ
⎛ z − ld ⎞ ⎞ ⎛ V τ σmD (z ) = σmo + ⎜2 f i ld − σmo⎟ exp ⎜ −ρ ⎟ ⎝ ⎠ ⎝ Vm rf rf ⎠
(8)
where σf(z) and σm(z) denote the fiber and matrix axial stresses at the z location, as shown in Fig. 2. Solving Eqs. (5) and (8) with the boundary conditions given by Eqs. (6) and (7), and the interface shear stress in the debonded region, the fiber and matrix axial stresses in the interface debonded region, i.e., 0 < z < ld, can be determined by Eqs. (9) and (10).
σfD (z )
2τ = T − i z, z ∈ (0, ld ) rf
σmD (z )
V τ = 2 f i z, z ∈ (0, ld ) Vm rf
τiD (z ) =
σfo = (9)
(10)
(12)
∂w ∂r
(13)
(14)
(15)
where Gm is the matrix shear modulus; and w is the axial displacement. Substituting the Eq. (14) into the Eq. (15), the interfacial shear stress τi(z), in the interface bonded region can be given by the Eq. (16).
τi (z ) =
Gm (wm − wf ) rf ln(R / rf )
dwm σ = m dz Em
(18)
F ∂wf (0) 1 − 4πrf ∂ld 2
∫0
ld
τi
∂v (z ) dz ∂ld
(26)
denotes the fiber load at the matrix cracking plane; where wf(0) denotes the fiber axial displacement on the matrix cracking plane; and v(z) denotes the relative displacement between the fiber and the matrix. The axial displacements of the fiber and matrix, i.e., wf(z) and wm(z), are given by Eqs. (27) and (28).
wf (z ) =
ld
∫∞
σfo r ⎛ l ⎞ T τ dz − f ⎜T − σfo − 2 d τi⎟ + (z − ld ) − i (z 2 − ld2 ) Ef ρEf ⎝ rf ⎠ Ef rf Ef (27)
where Ef and Em denote the fiber and matrix elastic modulus. Substituting Eqs. (16)–(18) into the Eq. (5), and applying the boundary conditions of Eqs. (6) and (7), the fiber and matrix axial stresses in the bonded region (ld < z) become
⎛ z − ld ⎞ ⎛ l ⎞ σfD (z ) = σfo + ⎜T − σfo − 2 d τi⎟ exp ⎜ −ρ ⎟ ⎝ ⎝ rf ⎠ rf ⎠
(25)
F(=πrf2σ/Vf)
where wf = w (rf , z ) and wm = w (R , z ) denote the fiber and the matrix axial displacement, respectively. (17)
σmU = σmo
ζd = −
(16)
dwf σ = f dz Ef
(24)
When matrix crack propagates to the fiber/matrix interface, it deflects along the fiber/matrix interface. There are two approaches to the problem of fiber/matrix interface debonding, namely, the shear strength approach and the fracture mechanics approach. The shear strength approach [24] is based upon a maximum shear stress criterion in which interface debonding occurs as the shear stress reaches the interface shear strength. On the other hand, the fracture mechanics approach [25] treats the interface debonding as a particular crack propagation problem in which interface debonding occurs as the strain energy release rate of the fiber/matrix interface achieves the interface debonding toughness. It has been proved that the fracture mechanics approach is preferred to the shear strength approach for interface debonding [26]. The fracture mechanics approach is adopted in the present analysis. The interface debonding criterion is given by Eq. (26) [25].
The matrix in the region rf < r < R only carries the shear stress, the stress-strain relation can be determined by the Eq. (15).
τrz = Gm
σfU = σfo
4. Interface debonding
The shear stress τrz is given by
rf τi (z ) r
(23)
The upstream region III as shown in Fig. 1 is so far away from the crack tip that the stress and strain fields are also uniform. The fiber and matrix have the same displacements and the fiber and matrix stresses are given by
Considering the equilibrium of the radius force acting on the differential element dz(dr)(rdθ) in the domain rf < r < R of the bonded matrix region (i.e., z≥ld), leads to the following differential equation.
τrz (r , z ) =
Em σ + Em (αc − αm ) ΔT Ec
3.2. Upstream stresses
The model can be further simplified by defining an effective radius R (rf < R < R ) such that the matrix axial load to be concentrated at R and the region between rf and R carries only the shear stress [3].
∂τrz τ + rz = 0 ∂r r
(22)
where Ec denotes the composite elastic modulus; αf, αm and αc denote the fiber, matrix and composite thermal expansion coefficient, respectively; and ΔT denotes the temperature difference between the fabricated temperature T0 and testing temperature T1 (ΔT=T1−T0).
(11)
⎛R ⎞ 2 ln Vf + Vm (3 − Vf ) ln ⎜ ⎟ = − ⎝ rf ⎠ 4Vm2
Ef σ + Ef (αc − αf ) ΔT Ec
σmo =
rf Vf
(21)
where ρ denotes the shear-lag model parameter, and
For the bonded region (ld < z) in the downstream Region I, the fiber and matrix axial stresses and the interfacial shear stress can be determined using the composite-cylinder model adopted by BHE [3]. The free body diagram of the composite-cylinder model is illustrated in Fig. 2, where the fiber closure traction T that causes interfacial debonding between the fiber and the matrix over a distance ld and the crack opening displacement v(0). The radius of the matrix cylinder is given by the Eq. (11).
R=
⎛ z − ld ⎞ ρ ⎛ Vm Em 2τ l ⎞ σ − i d ⎟ exp ⎜ −ρ ⎟ ⎜ ⎝ 2 ⎝ Vf Ec rf ⎠ rf ⎠
(20)
wm (z ) =
ld
∫∞
σmo rV ⎛ τ ⎞ V τi dz + f f ⎜T − σfo − 2 i ld⎟ + f (z 2 − ld2 ) Em ρVm Em ⎝ rf ⎠ Vm rf Em (28)
The relative displacement between the fiber and the matrix, i.e., v(z), is given by Eq. (29).
(19) 484
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
v (z ) = wf (z ) − wm (z ) ⎞ ⎛ r E l = ρV fE c E ⎜T − σfo − 2 rd τi⎟ + m m f⎝ f ⎠
T Ef
(l d − z ) −
Ec τ i rf Vm Em Ef
(ld2 − z 2 )
(29)
Substituting wf(z=0) and v(z) into Eq. (26), it leads to the form of Eq. (30).
⎛ Ec τ 2 ⎞ ⎛ r T2 Ec τi2 τ T⎞ r Tσ r τT i − i ⎟ ld + ⎜ f − f − f i − ζd⎟ = 0 ld2 + ⎜ Ef ⎠ 4Ec 2ρEf rf Vm Em Ef ⎠ ⎝ 4E f ⎝ ρVm Em Ef (30) Solving the Eq. (29), the interface debonded length ld is determined by Eq. (31).
ld =
rf ⎛ Vm Em 1⎞ T− ⎟ ⎜ 2 ⎝ Ec τi ρ⎠ ⎛ r f ⎞2 rf2 Vf Vm Ef Em T 2 ⎛ σ ⎞ rf Vm Em Ef ζd ⎟+ ⎜1 − ⎜ ⎟ − ⎝ Vf T ⎠ ⎝ 2ρ ⎠ 4Ec2 τi2 Ec τi2
−
(31)
5. Matrix cracking stress The energy relationship to evaluate the steady-state matrix cracking stress is expressed as [3] 1 2
∞
⎡ ⎣
∫−∞ ⎢ EVff (σfU − σfD )2 +
Vm (σ U Em m
2⎤ − σmD ) ⎥ dz + ⎦
1 2πR2Gm
∞
R
∫−∞ ∫r ( rf τri (z) )2πrdrdz f
⎛ 4V l ⎞ = Vm ζm + ⎜ rf d ⎟ ζd ⎝ f ⎠ (32) where ζm is the matrix fracture energy; and Gm is the matrix shear modulus. Substituting the fiber and matrix stresses, and the interface shear stress of Eqs. (9), (10), (19)–(21), (24) and (25), and the debonded length of Eq. (31) into the Eq. (32), the energy balance equation leads to the form of
η1 σ 2 + η2 σ + η3 = 0
(33)
where
η1 =
ld r VE + f f f Ec ρVm Em Ec
η2 = −
2Vf ld T 4Vf ld τi 2r V T + − f f Ec ρVm Em ρVm Em
2 4 ⎛τ ⎞ ⎛ V E ⎞ η3 = 3 ⎜ r i ⎟ ⎜ V Ef cE ⎟ ld3 + f ⎝ ⎠ ⎝ m m f⎠
+
(34a)
rf Vf Ec T 2 ρVm Em Ef
− Vm ζm −
−
Vf l d T 2 Ef
4Vf Ec τ i l d T ρVm Em Ef
+
−
(34b)
2Vf τ i ld2 rf E f
4Vf Ec τ i2 ld2 ρrf Vm Em Ef
4Vf l d ζd rf
T +
4Vf Ec τ i2 ρ2 Vm Em Ef
Fig. 3. (a) The matrix cracking stress versus fiber volume fraction; (b) the interface debonded length ld/rf versus fiber volume fraction; and (c) the broken fiber fraction versus fiber volume fraction.
ld (34c)
fiber Weibull modulus and fiber strength increase, the matrix cracking stress increases, and the interface debonded length and fiber broken fraction decrease. The comparisons of matrix cracking stress derived from the ACK model [2], Chiang model [4] and the present analysis have been investigated, as shown in Fig. 8. It was found that at the same interface shear stress, the matrix cracking stress predicted by the ACK model is the lowest, the Chiang model's results is the highest, and the present analysis lies between that of ACK model [2] and Chiang model [4] due to the consideration of synergistic effects of interface debonding and fibers failure.
6. Results and discussion The ceramic composite system of SiC/borosilicate is used for the case study and its material properties are given by [9]: Vf=40%, Ef=400 GPa, Em=63 GPa, rf=70 µm, ζm=8.92 J/m2, ζd=0.8 J/m2, τi=8 MPa, σc=2.0 GPa, and m=4. The effects of material properties, i.e., fiber volume fraction, interface shear stress, interface debonded energy, fiber Weibull modulus and fibers strength, on the matrix cracking stress, interface debonding and fibers failure have been analyzed, as shown in Figs. 3–7. It can be found that when the fiber volume fraction, interface shear stress and interface debonded energy increase, the matrix cracking stress and the fiber broken fraction increase, and the interface debonded length decreases; and when the
6.1. Effect of fiber volume fraction The matrix cracking stress σmc, interface debonded length ld/rf and broken fibers fraction versus fiber volume fraction curves are illu485
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
Fig. 4. (a) The matrix cracking stress versus interface shear stress; (b) the interface debonded length ld/rf versus interface shear stress; and (c) the broken fiber fraction versus interface shear stress.
Fig. 5. (a) The matrix cracking stress versus interface debonded energy ζd/ζm; (b) the interface debonded length ld/rf versus interface debonded energy ζd/ζm; and (c) the broken fiber fraction versus interface debonded energy ζd/ζm.
strated in Fig. 3. When the fiber volume fraction increases from 20% to 50%, the matrix cracking stress σmc increases from 93.6MPa to 324 MPa, as shown in Fig. 3(a); the interface debonded length ld/rf decreases from 5.5 to 2.2, as shown in Fig. 3(b); and the broken fibers fraction increases from 0.07% to 0.35%, as shown in Fig. 3(c). When the fiber volume fraction increases, the interface debonded length decreases due to more fibers carrying the applied load, leading to the higher matrix cracking stress and broken fibers fraction.
fiber broken fraction versus interface shear stress curves are illustrated in Fig. 4. When the interface shear stress increases from 1 MPa to 10 MPa, the matrix cracking stress σmc increases from 165 MPa to 240 MPa, as shown in Fig. 4(a); the interface debonded length ld/rf decreases from 9.5 to 2.5, as shown in Fig. 4(b); and the fiber broken fraction increases from 0.03% to 0.24%, as shown in Fig. 4(c). When the interface shear stress increases, the interface debonded length decreases due to the higher frictional resistance existed in the fiber/matrix interface, leading to the higher matrix cracking stress and broken fibers fraction.
6.2. Effect of interface shear stress The matrix cracking stress σmc, interface debonded length ld/rf and 486
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
Fig. 7. (a) The matrix cracking stress versus fiber strength; (b) the interface debonded length ld/rf versus fiber strength; and (c) the broken fiber fraction versus fiber strength.
Fig. 6. (a) The matrix cracking stress versus fiber Weibull modulus m; (b) the interface debonded length ld/rf versus fiber Weibull modulus m; and (c) the broken fiber fraction versus fiber Weibull modulus m.
cracking stress and broken fibers fraction.
6.3. Effect of interface debonded energy 6.4. Effect of fiber Weibull modulus The matrix cracking stress σmc, interface debonded length ld/rf and fiber broken fraction versus interface debonded energy ζd/ζm curves are illustrated in Fig. 5. When the interface debonded energy ζd/ζm increases from 0.1 to 0.4, the matrix cracking stress σmc increases from 234 MPa to 313 MPa, as shown in Fig. 5(a); the interface debonded length ld/rf decreases from 2.8 to 1.36, as shown in Fig. 5(b); and the fiber broken fraction increases from 0.21% to 0.94%, as shown in Fig. 5(c). When the interface debonded energy increases, the interface debonded length decreases due to the higher frictional resistance existed in the fiber/matrix interface, leading to the higher matrix
The matrix cracking stress σmc, interface debonded length ld/rf and fiber broken fraction versus fiber Weibull modulus curves are illustrated in Fig. 6. When the fiber Weibull modulus increases from 2 to 5, the matrix cracking stress σmc increases from 230 MPa to 235 MPa, as shown in Fig. 6(a); the interface debonded length ld/rf decreases from 2.86 to 2.81, as shown in Fig. 6(b); and the fiber broken fraction decreases from 2.4% to 0.06%, as shown in Fig. 6(c). When the fiber Weibull modulus increases, the fiber broken fraction decreases at the same applied stress, leading to the higher matrix cracking stress and the interface debonded length. 487
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
Fig. 8. (a) The matrix cracking stress versus interface shear stress curves corresponding to the present analysis, ACK model and Chiang model; (b) the interface debonded length ld/rf versus interface shear stress curves corresponding to the present analysis and Chiang model.
6.5. Effect of fiber strength The matrix cracking stress σmc, interface debonded length ld/rf and fiber broken fraction versus fiber strength σc curves are illustrated in Fig. 7. When the fiber strength σc increases from 1.5GPa to 2.0 GPa, the matrix cracking stress σmc increases from 232 MPa to 234 MPa, as shown in Fig. 7(a); the interface debonded length ld/rf decreases from 2.84 to 2.83, as shown in Fig. 7(b); and the fiber broken fraction decreases from 0.89% to 0.21%, as shown in Fig. 7(c). When the fiber strength increases, the fiber broken fraction decreases at the same applied stress, leading to the higher matrix cracking stress and the interface debonded length.
Fig. 9. (a) The experimental and theoretical matrix cracking stress versus fiber volume fraction; (b) the interface debonded length ld/rf versus fiber volume fraction; and (c) the broken fiber volume fraction versus fiber volume fraction corresponding to different interface debonded energy of SiC/borosilicate composite.
6.6. Comparisons with ACK model and Chiang model interface debonded toughness [4]. However, in the present analysis, the synergistic effects of interface debonding and fibers failure were both considered, leading to the decrease of interface debonded length compared with that of Chiang model, as shown in Fig. 8(b).
The matrix cracking stress versus interface shear stress curves corresponding to the present analysis, ACK model [2] and Chiang model [4] is illustrated in Fig. 8(a). At the same interface shear stress, the matrix cracking stress predicted by the ACK model is the lowest, and the Chiang model's result is the highest, and the present analysis lies between that of ACK model and Chiang model. For the ACK model and Chiang model, the interface debonding is considered in the matrix cracking with different interface debonding criterion, i.e., in the ACK model, the interface debonded length is determined as the load transfer length without considering interface debonded energy [2], and in the Chiang model, the fracture mechanics interface debonding criterion is adopted to determine the interface debonded length considering
7. Experimental comparisons The experimental and theoretical matrix cracking stress versus fiber volume fraction corresponding to different interface debonded energy ζd/ζm of SiC/borosilicate, SiC/LAS and C/borosilicate composites are illustrated in Figs. 9–11. The material properties of SiC/borosilicate, SiC/LAS and C/borosilicate composites are listed in Table 1. The 488
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
Fig. 11. (a) The experimental and theoretical matrix cracking stress versus fiber volume fraction; (b) the interface debonded length ld/rf versus fiber volume fraction; and (c) the broken fiber volume fraction versus fiber volume fraction corresponding to different interface debonded energy of C/borosilicate composite.
Fig. 10. (a) The experimental and theoretical matrix cracking stress versus fiber volume fraction; (b) the interface debonded length ld/rf versus fiber volume fraction; and (c) the broken fiber volume fraction versus fiber volume fraction corresponding to different interface debonded energy of SiC/LAS composite.
Table 1 The material properties of composite systems.
thermal residual stress due to thermal expansion mismatch between the fiber and the matrix needs to be considered when comparing with experimental data. The matrix cracking stress σmc is modified as, T σmc = σmc − σth
(35)
T where σmc is the theoretical matrix cracking calculated by the Eq. (33); and σth is the thermal residual stress. For SiC/borosilicate composite, the predicted results with the interface debonded energy range of ζd/ζm=0.05–0.4 agree with experimental data corresponding to the fiber volume fraction changing from 10% to 50%, as shown in Fig. 9(a). The interface debonded length ld/rf versus the fiber volume fraction curves corresponding to the interface debonded energy of ζd/ζm=0.05–0.4 are illustrated in Fig. 9(b), in
489
Items
SiC/borosilicate [9]
SiC/LAS [9]
C/borosilicate [9]
Ef/GPa Em/GPa rf/μm ζm/(J/m2) τi/MPa αf/(10−6/°C) αm/(10−6/°C) ΔT/(°C) σc/GPa m
400 63 70 8.92 8 3.5 3.2 ‒500 2.0 4
200 85 8 47 1 3.1 1.5 ‒675 2.0 4
380 63 4 8.92 10 0.1 3.2 ‒500 2.5 4
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
leading to the higher matrix cracking stress and broken fibers fraction. (2) When the interface shear stress and interface debonded energy increase, the interface debonded length decreases due to the higher frictional resistance existed in the fiber/matrix interface, leading to the higher matrix cracking stress and broken fibers fraction. (3) When the fiber Weibull modulus and fiber strength increases, the fiber broken fraction decreases at the same applied stress, leading to the higher matrix cracking stress and the interface debonded length. (4) At the same interface shear stress, the matrix cracking stress predicted by the ACK model is the lowest, the Chiang model's results is the highest, and the present analysis lies between that of ACK model [2] and Chiang model [4] due to the consideration of synergistic effects of interface debonding and fibers failure.
which the interface debonded length ld/rf decreases with increasing interface debonded length at the same fiber volume content. The broken fibers fraction versus fiber volume fraction curves corresponding to the interface debonded energy of ζd/ζm=0.05–0.4 are illustrated in Fig. 9(c), in which the broken fiber fraction increases with increasing interface debonded energy at the same fiber volume fraction. For SiC/LAS composite, the matrix cracking stress with the interface debonded energy range of ζd/ζm=0.02–0.06 agrees with the experimental data corresponding to the fiber volume fraction changing from 30% to 50%, as shown in Fig. 10(a). The interface debonded length ld/rf versus the fiber volume fraction curves corresponding to the interface debonded energy of ζd/ζm=0.02–0.06 are illustrated in Fig. 10(b), in which the interface debonded length ld/rf decreases with increasing interface debonded length at the same fiber volume content. The broken fibers fraction versus fiber volume fraction curves corresponding to the interface debonded energy of ζd/ζm=0.02–0.06 are illustrated in Fig. 10(c), in which the broken fiber fraction increases with increasing interface debonded energy at the same fiber volume fraction. For C/borosilicate composite, the predicted matrix cracking stress with the interface debonded energy ramge of ζd/ζm=0.005–0.015 agrees with the experimental data corresponding to the fiber volume fraction changing from 30% to 55%, as shown in Fig. 11(a). The interface debonded length ld/rf versus the fiber volume fraction curves corresponding to the interface debonded energy of ζd/ζm=0.005–0.015 are illustrated in Fig. 11(b), in which the interface debonded length ld/ rf decreases with increasing interface debonded length at the same fiber volume content. The broken fibers fraction versus fiber volume fraction curves corresponding to the interface debonded energy of ζd/ ζm=0.005–0.015 are illustrated in Fig. 11(c), in which the broken fiber fraction increases with increasing interface debonded energy at the same fiber volume fraction. From Figs. 9–11, it can be found that the fiber/matrix interface is with strong bonding in SiC/borosilicate composite, and weak bonding in SiC/LAS and C/borosilicate composites.
Acknowledgements The work reported here is supported by the Natural Science Fund of Jiangsu Province (Grant no. BK20140813), and the Fundamental Research Funds for the Central Universities (Grant no. NS2016070). The author would also thank the anonymous reviewers and the editor for their valuable comments on an earlier version of the paper. References [1] R.R. Naslain, Compos. Sci. Technol. 64 (2004) 155–170. [2] J. Aveston, G.A. Cooper, A. Kelly. In: Proceedings of the Conference on The Properties of Fiber Composites, National Physical Laboratory, IPC Science and Technology Press, Guildford, 1971, pp. 15–26. [3] B. Budiansky, J.W. Hutchinson, A.G. Evans, J. Mech. Phys. Solids 34 (1986) 167–189. [4] Y.C. Chiang, Eng. Fract. Mech. 65 (2000) 15–28. [5] D.B. Marshall, B.N. Cox, A.G. Evans, Acta Metall. 33 (1985) 2013–2021. [6] L.N. McCartney, Proc. R. Soc. Lond. 409 (1987) 329–350. [7] J. Aveston, A. Kelly, J. Mater. Sci. 8 (1973) 352–362. [8] V.P. Rajan, F.W. Zok, J. Mech. Phys. Solids 73 (2014) 3–21. [9] M.W. Barsoum, P. Kangutkar, A.S.D. Wang, Compos. Sci. Technol. 44 (1993) 257–269. [10] M.D. Thouless, A.G. Evans, Acta Metall. 36 (1988) 517–522. [11] H.C. Cao, M.D. Thouless, J. Am. Ceram. Soc. 73 (1990) 2091–2094. [12] M. Sutcu, Acta Metall. 37 (1989) 651–661. [13] H.R. Schwietert, P.S. Steif, J. Mech. Phys. Solids 38 (1990) (1990) 325–343. [14] W.A. Curtin, J. Am. Ceram. Soc. 74 (1991) 2837–2845. [15] Y. Weitsman, H. Zhu, J. Mech. Phys. Solids 41 (1993) 351–388. [16] F. Hild, J.M. Domergue, F.A. Leckie, A.G. Evans, Int. J. Solids Struct. 31 (1994) 1035–1045. [17] J.P. Solti, S. Mall, D.D. Robertson, Compos. Sci. Technol. 54 (1995) 55–66. [18] C.D. Cho, J. Mech. Sci. Technol. 10 (1996) 49–56. [19] R. Paar, J.L. Valles, R. Danzer, Mater. Sci. Eng. A 250 (1998) 209–216. [20] K. Liao, R.L. Reifsnider, Int. J. Fract. 106 (2000) 95–115. [21] S.J. Zhou, W.A. Curtin, Acta Metall. Mater. 43 (1995) 3093–3104. [22] R.E. Dutton, N.J. Pagano, R.Y. Kim, T.A. Parthasarathy, J. Am. Ceram. Soc. 83 (2000) 166–174. [23] Z.H. Xia, W.A. Curtin, Acta Mater. 48 (2000) 4879–4892. [24] C.H. Hsueh, Acta Mater. 44 (1996) 2211–2216. [25] Y.C. Gao, Y.W. Mai, B. Cotterell, J. Appl. Math. Phys. 39 (1988) 550–572. [26] Y.J. Sun, R.N. Singh, Acta Mater. 46 (1998) 1657–1667.
8. Conclusions In this paper, the synergistic effects of fiber debonding and fracture on matrix cracking stress of fiber-reinforced CMCs have been investigated using the energy balance approach. The experimental matrix cracking stress of three different CMCs, i.e., SiC/borosilicate, SiC/LAS, and C/borosilicate, with different fiber volume fraction have been predicted. It was found that the fiber/matrix interface possesses strong bonding in SiC/borosilicate composite, and weak bonding in SiC/LAS and C/borosilicate composites. The effects of fiber volume fraction, interface shear stress, interface debonded energy, fiber Weibull modulus, and fiber strength on matrix cracking stress, interface debonded length and fiber broken fraction have been analyzed. (1) When the fiber volume fraction increases, the interface debonded length decreases due to more fibers carrying the applied load,
490
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Synergistic effects of fiber debonding and fracture on matrix cracking in fiber-reinforced ceramic-matrix composites
crossmark
Li Longbiao College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, No. 29 Yudao St., Nanjing 210016, PR China
A R T I C L E I N F O
A BS T RAC T
Keywords: Ceramic-matrix composites (CMCs) Matrix cracking Interface debonding Fiber failure
In this paper, the synergistic effects of fiber debonding and fracture on matrix cracking stress of fiber-reinforced ceramic-matrix composites (CMCs) have been investigated using the energy balance approach. The shear-lag model cooperated with fiber/matrix interface debonding criterion and fiber fracture model has been adopted to analyze stress distribution in CMCs. The relationships between matrix cracking stress, interface debonding and slipping, and fiber fracture have been established. The effects of fiber volume fraction, interface shear stress, interface debonded energy, fiber Weibull modulus, and fiber strength on matrix cracking stress, interface debonded length and fiber broken fraction have been analyzed. The experimental matrix cracking stress of three different CMCs, i.e., SiC/borosilicate, SiC/LAS, and C/borosilicate, with different fiber volume fraction have been predicted.
1. Introduction Ceramic materials possess high strength and modulus at elevated temperature. But their use as structural components is severely limited because of their brittleness. Continuous fiber-reinforced ceramicmatrix composites, by incorporating fibers in ceramic matrices, however, not only exploit their attractive high-temperature strength but also reduce the propensity for catastrophic failure. These materials have already been implemented on some aero engines’ components [1]. The CMCs exhibit distinct behaviors at stresses above and below the matrix cracking stress, which is associated with the onset of matrix cracking and with the formation of hysteresis loops that results from matrix cracking and frictional slipping of the fibers bridging matrix cracks. In the environmentally-stable fibers and fiber coating in oxidizing environments, the matrix cracking stress has long been considered the maximum allowable design stress for CMCs in the real applications involving oxidizing environments. Many researchers performed experimental and theoretical investigations on matrix cracking of fiber-reinforced CMCs. For analytical modeling, the energy balance approach developed by Aveston, Cooper and Kelly (ACK) [2], Budiansky, Hutchinson and Evans (BHE) [3], and Chiang [4], and the fracture mechanics approach proposed by Marshall, Cox and Evans (MCE) [5], and McCartney [6] have been used to investigate the matrix cracking stress. The analytical results show that the matrix cracking stress was closely related with the interface friction stress. The composite with the higher interface shear stress results in the higher matrix cracking stress. When the fiber/
matrix interface is weakly bonding, the BHE model will reduce to the ACK model, and when the interface is strongly bonding, the matrix cracking stress predicted by the BHE model is the same with that of Aveston and Kelly [7]. Rajan and Zok [8] investigate the mechanics of a fully bridged steady-state matrix cracking in unidirectional CMCs under shear loading. However, the models mentioned above do not consider the synergistic effects of fiber debonding and fracture on matrix cracking stress in CMCs. During the process of matrix cracking in CMCs, the fiber/matrix interfacial debonding occurs due to the fiber-matrix relative displacement above the matrix cracking plane, and fibers failure also occurs due to the statistical properties of the fibers strength. In the present analysis, the shear-lag model is adopted to analyze the micro-stress field of the damaged composite, including the fiber, matrix and fiber/ matrix interface shear stress in the interface debonded and bonded region. The Global Load Sharing criterion (GLS) and the fracture mechanics approach were used to determine the broken fibers fraction and the interface debonded length during matrix cracking. The effects of interface debonding and slipping, and fibers fracture have been taken into consideration to solve the matrix cracking stress. The influence of material properties, i.e., fiber volume fraction, interface shear stress, interface debonded energy, fiber Weibull modulus, and fiber strength on matrix cracking stress, interface debonded length and fiber broken fraction have been analyzed.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.msea.2016.11.077 Received 20 September 2016; Received in revised form 14 October 2016; Accepted 22 November 2016 Available online 23 November 2016 0921-5093/ © 2016 Published by Elsevier B.V.
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
2. Materials and experimental procedures The three unidirectional fiber-reinforced CMCs for matrix cracking were provided by Barsoum et al. [9], including (a) the SiC/lithium/ aluminosilicate (SiC/LAS) system, which comprises a SiC fiber (Nicalon, Nippon Carbon Co., Tokyo Japan) and a LAS matrix; (b) the carbon/borosilicate system, which is made of HMU carbon fibers (HMU Hercules carbon fiber) embedded in a borosilicate glass matrix; and (c) the SiC/borosilicate system, consisting of a SiC monofilament (SCS-6 SiC Fiber, Textron Specialty Materials, Lowell, Massachusetts, USA) embedded in the same borosilicate glass matrix. The variation in fiber volume fraction is achieved by controlling the amount of glass powder used in the preparation of the polymer slurry. Three-point-bend test specimens were performed in the form of flat bars having the dimensions of 3×0.5×0.18 cm3 for SiC/LAS, 3×0.5×0.2 cm3 for C/borosilicate, and 6×0.5×0.2 cm3 for SiC/borosilicate. The pin-support span of the three-point bend fixture was 2.54 cm for the shorter bars and 5.2 cm for the longer bars. During the three-point-bend testing, the maximum beam deflection was measured with an extensometer mounted at the beam mid-span; at the same time, the electrical resistance of the gold film sputtered on the polished specimen surface was measured with a digital Ohm meter. The deflection and electrical resistance were recorded as a continuous function of loading and were then used to determine the onset of the matrix cracking stress.
Fig. 1. The schematic of crack-tip and interface debonding.
Substituting Eqs. (2) and (3) into the Eq. (1), it leads into the form of
⎡ ⎛ T ⎞m +1⎤ ⎫ ⎛ σ ⎞m +1 ⎧ ⎪ ⎪ σ ⎬ = T ⎜ c⎟ ⎨ 1 − exp ⎢ −⎜ ⎟ ⎥ ⎪ ⎪ ⎝T ⎠ ⎩ Vf ⎣⎢ ⎝ σc ⎠ ⎥⎦ ⎭
Using the Eq. (4), the stress T carried by intact fibers at the matrix cracking plane can be determined. Substituting the intact fiber stress T into the Eq. (2), the relationship between the fiber failure probability and applied stress can be determined.
3. Stress analysis When fibers break during matrix cracking, the loads dropped by the broken fibers must be transferred to the intact fibers in the crosssection. Two dominant failure criterions are present in the literatures for modeling fibers failure, i.e., Global Load Sharing criterion (GLS) and Local Load Sharing criterions (LLS). The GLS criterion assumes that the load from any one fiber is transferred equally to all other intact fibers in the same cross-section plane. The GLS assumption neglects any local stress concentrations in the neighborhood of existing breaks, and is expected to be accurate when the interfacial shear stress is sufficiently low. Models that include GLS explicitly have been developed, which includes Thouless and Evans [10], Cao and Thouless [11], Sutcu [12], Schwietert and Steif [13], Curtin [14], Weitsman and Zhu [15], Hild et al. [16], Solti et al. [17], Cho [18], Paar et al. [19], Liao and Reifsnider [20], and so on. The LLS assumes that the load from the broken fiber is transferred to the neighborhood intact fibers, and is expected to be accurate when the interface shear stress is sufficiently high. Models that include LLS explicitly have been developed, which includes Zhou and Curtin [21], Dutton et al. [22], Xia and Curtin [23], and so on. The two-parameter Weibull model is adopted to describe the fiber strength distribution, and the Global Load Sharing (GLS) assumption is used to determine the load carried by intact and fracture fibers [14].
σ = T [1 − P (T )] + Tb P (T ) Vf
3.1. Downstream stresses The composite with fiber volume fraction Vf is loaded by a remote uniform stress σ normal to a long crack plane, as shown in Fig. 1. The unit cell in the downstream Region I contained a single fiber surrounded by a hollow cylinder of matrix is extracted from the ceramic composite system, as shown in Fig. 2. The fiber radius is rf, and the matrix radius is R (R=rf/Vf1/2). The length of the unit cell is half matrix crack spacing lc/2, and the interface debonded length is ld. In the debonded region, the interface is resisted by τi. For the debonded region in Region I, the force equilibrium equation of the fiber is given by Eq. (5) [3].
dσf (z ) 2τ (z ) =− i dz rf
σf (z = 0) = T
(6)
σm (z = 0) = 0
(7)
The total axial stresses in Region I satisfy the Eq. (8).
(1)
(2)
where m denotes the fiber Weibull modulus; and σc denotes the fiber characteristic strength. The load carried by broken fibers is determined by the Eq. (3).
⎡ ⎛ σ ⎞m +1 1 − P (T ) ⎤ ⎥ Tb = T ⎢ ⎜ c ⎟ − ⎢⎣ ⎝ T ⎠ P (T ) ⎥⎦
(5)
The boundary conditions of the fiber and matrix axial stresses at the crack plane (i.e., z=0) are given by
where Vf denotes the fiber volume fraction; T denotes the load carried by intact fibers; Tb denotes the load carried by broken fibers; and P(T) denotes the fiber failure probability.
⎡ ⎛ T ⎞m +1⎤ P (T ) = 1 − exp ⎢ −⎜ ⎟ ⎥ ⎢⎣ ⎝ σc ⎠ ⎥⎦
(4)
(3)
Fig. 2. The schematic of shear-lag model considering interface debonding.
483
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
Vf σf (z ) + Vm σm (z ) = σ
⎛ z − ld ⎞ ⎞ ⎛ V τ σmD (z ) = σmo + ⎜2 f i ld − σmo⎟ exp ⎜ −ρ ⎟ ⎝ ⎠ ⎝ Vm rf rf ⎠
(8)
where σf(z) and σm(z) denote the fiber and matrix axial stresses at the z location, as shown in Fig. 2. Solving Eqs. (5) and (8) with the boundary conditions given by Eqs. (6) and (7), and the interface shear stress in the debonded region, the fiber and matrix axial stresses in the interface debonded region, i.e., 0 < z < ld, can be determined by Eqs. (9) and (10).
σfD (z )
2τ = T − i z, z ∈ (0, ld ) rf
σmD (z )
V τ = 2 f i z, z ∈ (0, ld ) Vm rf
τiD (z ) =
σfo = (9)
(10)
(12)
∂w ∂r
(13)
(14)
(15)
where Gm is the matrix shear modulus; and w is the axial displacement. Substituting the Eq. (14) into the Eq. (15), the interfacial shear stress τi(z), in the interface bonded region can be given by the Eq. (16).
τi (z ) =
Gm (wm − wf ) rf ln(R / rf )
dwm σ = m dz Em
(18)
F ∂wf (0) 1 − 4πrf ∂ld 2
∫0
ld
τi
∂v (z ) dz ∂ld
(26)
denotes the fiber load at the matrix cracking plane; where wf(0) denotes the fiber axial displacement on the matrix cracking plane; and v(z) denotes the relative displacement between the fiber and the matrix. The axial displacements of the fiber and matrix, i.e., wf(z) and wm(z), are given by Eqs. (27) and (28).
wf (z ) =
ld
∫∞
σfo r ⎛ l ⎞ T τ dz − f ⎜T − σfo − 2 d τi⎟ + (z − ld ) − i (z 2 − ld2 ) Ef ρEf ⎝ rf ⎠ Ef rf Ef (27)
where Ef and Em denote the fiber and matrix elastic modulus. Substituting Eqs. (16)–(18) into the Eq. (5), and applying the boundary conditions of Eqs. (6) and (7), the fiber and matrix axial stresses in the bonded region (ld < z) become
⎛ z − ld ⎞ ⎛ l ⎞ σfD (z ) = σfo + ⎜T − σfo − 2 d τi⎟ exp ⎜ −ρ ⎟ ⎝ ⎝ rf ⎠ rf ⎠
(25)
F(=πrf2σ/Vf)
where wf = w (rf , z ) and wm = w (R , z ) denote the fiber and the matrix axial displacement, respectively. (17)
σmU = σmo
ζd = −
(16)
dwf σ = f dz Ef
(24)
When matrix crack propagates to the fiber/matrix interface, it deflects along the fiber/matrix interface. There are two approaches to the problem of fiber/matrix interface debonding, namely, the shear strength approach and the fracture mechanics approach. The shear strength approach [24] is based upon a maximum shear stress criterion in which interface debonding occurs as the shear stress reaches the interface shear strength. On the other hand, the fracture mechanics approach [25] treats the interface debonding as a particular crack propagation problem in which interface debonding occurs as the strain energy release rate of the fiber/matrix interface achieves the interface debonding toughness. It has been proved that the fracture mechanics approach is preferred to the shear strength approach for interface debonding [26]. The fracture mechanics approach is adopted in the present analysis. The interface debonding criterion is given by Eq. (26) [25].
The matrix in the region rf < r < R only carries the shear stress, the stress-strain relation can be determined by the Eq. (15).
τrz = Gm
σfU = σfo
4. Interface debonding
The shear stress τrz is given by
rf τi (z ) r
(23)
The upstream region III as shown in Fig. 1 is so far away from the crack tip that the stress and strain fields are also uniform. The fiber and matrix have the same displacements and the fiber and matrix stresses are given by
Considering the equilibrium of the radius force acting on the differential element dz(dr)(rdθ) in the domain rf < r < R of the bonded matrix region (i.e., z≥ld), leads to the following differential equation.
τrz (r , z ) =
Em σ + Em (αc − αm ) ΔT Ec
3.2. Upstream stresses
The model can be further simplified by defining an effective radius R (rf < R < R ) such that the matrix axial load to be concentrated at R and the region between rf and R carries only the shear stress [3].
∂τrz τ + rz = 0 ∂r r
(22)
where Ec denotes the composite elastic modulus; αf, αm and αc denote the fiber, matrix and composite thermal expansion coefficient, respectively; and ΔT denotes the temperature difference between the fabricated temperature T0 and testing temperature T1 (ΔT=T1−T0).
(11)
⎛R ⎞ 2 ln Vf + Vm (3 − Vf ) ln ⎜ ⎟ = − ⎝ rf ⎠ 4Vm2
Ef σ + Ef (αc − αf ) ΔT Ec
σmo =
rf Vf
(21)
where ρ denotes the shear-lag model parameter, and
For the bonded region (ld < z) in the downstream Region I, the fiber and matrix axial stresses and the interfacial shear stress can be determined using the composite-cylinder model adopted by BHE [3]. The free body diagram of the composite-cylinder model is illustrated in Fig. 2, where the fiber closure traction T that causes interfacial debonding between the fiber and the matrix over a distance ld and the crack opening displacement v(0). The radius of the matrix cylinder is given by the Eq. (11).
R=
⎛ z − ld ⎞ ρ ⎛ Vm Em 2τ l ⎞ σ − i d ⎟ exp ⎜ −ρ ⎟ ⎜ ⎝ 2 ⎝ Vf Ec rf ⎠ rf ⎠
(20)
wm (z ) =
ld
∫∞
σmo rV ⎛ τ ⎞ V τi dz + f f ⎜T − σfo − 2 i ld⎟ + f (z 2 − ld2 ) Em ρVm Em ⎝ rf ⎠ Vm rf Em (28)
The relative displacement between the fiber and the matrix, i.e., v(z), is given by Eq. (29).
(19) 484
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
v (z ) = wf (z ) − wm (z ) ⎞ ⎛ r E l = ρV fE c E ⎜T − σfo − 2 rd τi⎟ + m m f⎝ f ⎠
T Ef
(l d − z ) −
Ec τ i rf Vm Em Ef
(ld2 − z 2 )
(29)
Substituting wf(z=0) and v(z) into Eq. (26), it leads to the form of Eq. (30).
⎛ Ec τ 2 ⎞ ⎛ r T2 Ec τi2 τ T⎞ r Tσ r τT i − i ⎟ ld + ⎜ f − f − f i − ζd⎟ = 0 ld2 + ⎜ Ef ⎠ 4Ec 2ρEf rf Vm Em Ef ⎠ ⎝ 4E f ⎝ ρVm Em Ef (30) Solving the Eq. (29), the interface debonded length ld is determined by Eq. (31).
ld =
rf ⎛ Vm Em 1⎞ T− ⎟ ⎜ 2 ⎝ Ec τi ρ⎠ ⎛ r f ⎞2 rf2 Vf Vm Ef Em T 2 ⎛ σ ⎞ rf Vm Em Ef ζd ⎟+ ⎜1 − ⎜ ⎟ − ⎝ Vf T ⎠ ⎝ 2ρ ⎠ 4Ec2 τi2 Ec τi2
−
(31)
5. Matrix cracking stress The energy relationship to evaluate the steady-state matrix cracking stress is expressed as [3] 1 2
∞
⎡ ⎣
∫−∞ ⎢ EVff (σfU − σfD )2 +
Vm (σ U Em m
2⎤ − σmD ) ⎥ dz + ⎦
1 2πR2Gm
∞
R
∫−∞ ∫r ( rf τri (z) )2πrdrdz f
⎛ 4V l ⎞ = Vm ζm + ⎜ rf d ⎟ ζd ⎝ f ⎠ (32) where ζm is the matrix fracture energy; and Gm is the matrix shear modulus. Substituting the fiber and matrix stresses, and the interface shear stress of Eqs. (9), (10), (19)–(21), (24) and (25), and the debonded length of Eq. (31) into the Eq. (32), the energy balance equation leads to the form of
η1 σ 2 + η2 σ + η3 = 0
(33)
where
η1 =
ld r VE + f f f Ec ρVm Em Ec
η2 = −
2Vf ld T 4Vf ld τi 2r V T + − f f Ec ρVm Em ρVm Em
2 4 ⎛τ ⎞ ⎛ V E ⎞ η3 = 3 ⎜ r i ⎟ ⎜ V Ef cE ⎟ ld3 + f ⎝ ⎠ ⎝ m m f⎠
+
(34a)
rf Vf Ec T 2 ρVm Em Ef
− Vm ζm −
−
Vf l d T 2 Ef
4Vf Ec τ i l d T ρVm Em Ef
+
−
(34b)
2Vf τ i ld2 rf E f
4Vf Ec τ i2 ld2 ρrf Vm Em Ef
4Vf l d ζd rf
T +
4Vf Ec τ i2 ρ2 Vm Em Ef
Fig. 3. (a) The matrix cracking stress versus fiber volume fraction; (b) the interface debonded length ld/rf versus fiber volume fraction; and (c) the broken fiber fraction versus fiber volume fraction.
ld (34c)
fiber Weibull modulus and fiber strength increase, the matrix cracking stress increases, and the interface debonded length and fiber broken fraction decrease. The comparisons of matrix cracking stress derived from the ACK model [2], Chiang model [4] and the present analysis have been investigated, as shown in Fig. 8. It was found that at the same interface shear stress, the matrix cracking stress predicted by the ACK model is the lowest, the Chiang model's results is the highest, and the present analysis lies between that of ACK model [2] and Chiang model [4] due to the consideration of synergistic effects of interface debonding and fibers failure.
6. Results and discussion The ceramic composite system of SiC/borosilicate is used for the case study and its material properties are given by [9]: Vf=40%, Ef=400 GPa, Em=63 GPa, rf=70 µm, ζm=8.92 J/m2, ζd=0.8 J/m2, τi=8 MPa, σc=2.0 GPa, and m=4. The effects of material properties, i.e., fiber volume fraction, interface shear stress, interface debonded energy, fiber Weibull modulus and fibers strength, on the matrix cracking stress, interface debonding and fibers failure have been analyzed, as shown in Figs. 3–7. It can be found that when the fiber volume fraction, interface shear stress and interface debonded energy increase, the matrix cracking stress and the fiber broken fraction increase, and the interface debonded length decreases; and when the
6.1. Effect of fiber volume fraction The matrix cracking stress σmc, interface debonded length ld/rf and broken fibers fraction versus fiber volume fraction curves are illu485
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
Fig. 4. (a) The matrix cracking stress versus interface shear stress; (b) the interface debonded length ld/rf versus interface shear stress; and (c) the broken fiber fraction versus interface shear stress.
Fig. 5. (a) The matrix cracking stress versus interface debonded energy ζd/ζm; (b) the interface debonded length ld/rf versus interface debonded energy ζd/ζm; and (c) the broken fiber fraction versus interface debonded energy ζd/ζm.
strated in Fig. 3. When the fiber volume fraction increases from 20% to 50%, the matrix cracking stress σmc increases from 93.6MPa to 324 MPa, as shown in Fig. 3(a); the interface debonded length ld/rf decreases from 5.5 to 2.2, as shown in Fig. 3(b); and the broken fibers fraction increases from 0.07% to 0.35%, as shown in Fig. 3(c). When the fiber volume fraction increases, the interface debonded length decreases due to more fibers carrying the applied load, leading to the higher matrix cracking stress and broken fibers fraction.
fiber broken fraction versus interface shear stress curves are illustrated in Fig. 4. When the interface shear stress increases from 1 MPa to 10 MPa, the matrix cracking stress σmc increases from 165 MPa to 240 MPa, as shown in Fig. 4(a); the interface debonded length ld/rf decreases from 9.5 to 2.5, as shown in Fig. 4(b); and the fiber broken fraction increases from 0.03% to 0.24%, as shown in Fig. 4(c). When the interface shear stress increases, the interface debonded length decreases due to the higher frictional resistance existed in the fiber/matrix interface, leading to the higher matrix cracking stress and broken fibers fraction.
6.2. Effect of interface shear stress The matrix cracking stress σmc, interface debonded length ld/rf and 486
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
Fig. 7. (a) The matrix cracking stress versus fiber strength; (b) the interface debonded length ld/rf versus fiber strength; and (c) the broken fiber fraction versus fiber strength.
Fig. 6. (a) The matrix cracking stress versus fiber Weibull modulus m; (b) the interface debonded length ld/rf versus fiber Weibull modulus m; and (c) the broken fiber fraction versus fiber Weibull modulus m.
cracking stress and broken fibers fraction.
6.3. Effect of interface debonded energy 6.4. Effect of fiber Weibull modulus The matrix cracking stress σmc, interface debonded length ld/rf and fiber broken fraction versus interface debonded energy ζd/ζm curves are illustrated in Fig. 5. When the interface debonded energy ζd/ζm increases from 0.1 to 0.4, the matrix cracking stress σmc increases from 234 MPa to 313 MPa, as shown in Fig. 5(a); the interface debonded length ld/rf decreases from 2.8 to 1.36, as shown in Fig. 5(b); and the fiber broken fraction increases from 0.21% to 0.94%, as shown in Fig. 5(c). When the interface debonded energy increases, the interface debonded length decreases due to the higher frictional resistance existed in the fiber/matrix interface, leading to the higher matrix
The matrix cracking stress σmc, interface debonded length ld/rf and fiber broken fraction versus fiber Weibull modulus curves are illustrated in Fig. 6. When the fiber Weibull modulus increases from 2 to 5, the matrix cracking stress σmc increases from 230 MPa to 235 MPa, as shown in Fig. 6(a); the interface debonded length ld/rf decreases from 2.86 to 2.81, as shown in Fig. 6(b); and the fiber broken fraction decreases from 2.4% to 0.06%, as shown in Fig. 6(c). When the fiber Weibull modulus increases, the fiber broken fraction decreases at the same applied stress, leading to the higher matrix cracking stress and the interface debonded length. 487
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
Fig. 8. (a) The matrix cracking stress versus interface shear stress curves corresponding to the present analysis, ACK model and Chiang model; (b) the interface debonded length ld/rf versus interface shear stress curves corresponding to the present analysis and Chiang model.
6.5. Effect of fiber strength The matrix cracking stress σmc, interface debonded length ld/rf and fiber broken fraction versus fiber strength σc curves are illustrated in Fig. 7. When the fiber strength σc increases from 1.5GPa to 2.0 GPa, the matrix cracking stress σmc increases from 232 MPa to 234 MPa, as shown in Fig. 7(a); the interface debonded length ld/rf decreases from 2.84 to 2.83, as shown in Fig. 7(b); and the fiber broken fraction decreases from 0.89% to 0.21%, as shown in Fig. 7(c). When the fiber strength increases, the fiber broken fraction decreases at the same applied stress, leading to the higher matrix cracking stress and the interface debonded length.
Fig. 9. (a) The experimental and theoretical matrix cracking stress versus fiber volume fraction; (b) the interface debonded length ld/rf versus fiber volume fraction; and (c) the broken fiber volume fraction versus fiber volume fraction corresponding to different interface debonded energy of SiC/borosilicate composite.
6.6. Comparisons with ACK model and Chiang model interface debonded toughness [4]. However, in the present analysis, the synergistic effects of interface debonding and fibers failure were both considered, leading to the decrease of interface debonded length compared with that of Chiang model, as shown in Fig. 8(b).
The matrix cracking stress versus interface shear stress curves corresponding to the present analysis, ACK model [2] and Chiang model [4] is illustrated in Fig. 8(a). At the same interface shear stress, the matrix cracking stress predicted by the ACK model is the lowest, and the Chiang model's result is the highest, and the present analysis lies between that of ACK model and Chiang model. For the ACK model and Chiang model, the interface debonding is considered in the matrix cracking with different interface debonding criterion, i.e., in the ACK model, the interface debonded length is determined as the load transfer length without considering interface debonded energy [2], and in the Chiang model, the fracture mechanics interface debonding criterion is adopted to determine the interface debonded length considering
7. Experimental comparisons The experimental and theoretical matrix cracking stress versus fiber volume fraction corresponding to different interface debonded energy ζd/ζm of SiC/borosilicate, SiC/LAS and C/borosilicate composites are illustrated in Figs. 9–11. The material properties of SiC/borosilicate, SiC/LAS and C/borosilicate composites are listed in Table 1. The 488
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
Fig. 11. (a) The experimental and theoretical matrix cracking stress versus fiber volume fraction; (b) the interface debonded length ld/rf versus fiber volume fraction; and (c) the broken fiber volume fraction versus fiber volume fraction corresponding to different interface debonded energy of C/borosilicate composite.
Fig. 10. (a) The experimental and theoretical matrix cracking stress versus fiber volume fraction; (b) the interface debonded length ld/rf versus fiber volume fraction; and (c) the broken fiber volume fraction versus fiber volume fraction corresponding to different interface debonded energy of SiC/LAS composite.
Table 1 The material properties of composite systems.
thermal residual stress due to thermal expansion mismatch between the fiber and the matrix needs to be considered when comparing with experimental data. The matrix cracking stress σmc is modified as, T σmc = σmc − σth
(35)
T where σmc is the theoretical matrix cracking calculated by the Eq. (33); and σth is the thermal residual stress. For SiC/borosilicate composite, the predicted results with the interface debonded energy range of ζd/ζm=0.05–0.4 agree with experimental data corresponding to the fiber volume fraction changing from 10% to 50%, as shown in Fig. 9(a). The interface debonded length ld/rf versus the fiber volume fraction curves corresponding to the interface debonded energy of ζd/ζm=0.05–0.4 are illustrated in Fig. 9(b), in
489
Items
SiC/borosilicate [9]
SiC/LAS [9]
C/borosilicate [9]
Ef/GPa Em/GPa rf/μm ζm/(J/m2) τi/MPa αf/(10−6/°C) αm/(10−6/°C) ΔT/(°C) σc/GPa m
400 63 70 8.92 8 3.5 3.2 ‒500 2.0 4
200 85 8 47 1 3.1 1.5 ‒675 2.0 4
380 63 4 8.92 10 0.1 3.2 ‒500 2.5 4
Materials Science & Engineering A 682 (2017) 482–490
L. Longbiao
leading to the higher matrix cracking stress and broken fibers fraction. (2) When the interface shear stress and interface debonded energy increase, the interface debonded length decreases due to the higher frictional resistance existed in the fiber/matrix interface, leading to the higher matrix cracking stress and broken fibers fraction. (3) When the fiber Weibull modulus and fiber strength increases, the fiber broken fraction decreases at the same applied stress, leading to the higher matrix cracking stress and the interface debonded length. (4) At the same interface shear stress, the matrix cracking stress predicted by the ACK model is the lowest, the Chiang model's results is the highest, and the present analysis lies between that of ACK model [2] and Chiang model [4] due to the consideration of synergistic effects of interface debonding and fibers failure.
which the interface debonded length ld/rf decreases with increasing interface debonded length at the same fiber volume content. The broken fibers fraction versus fiber volume fraction curves corresponding to the interface debonded energy of ζd/ζm=0.05–0.4 are illustrated in Fig. 9(c), in which the broken fiber fraction increases with increasing interface debonded energy at the same fiber volume fraction. For SiC/LAS composite, the matrix cracking stress with the interface debonded energy range of ζd/ζm=0.02–0.06 agrees with the experimental data corresponding to the fiber volume fraction changing from 30% to 50%, as shown in Fig. 10(a). The interface debonded length ld/rf versus the fiber volume fraction curves corresponding to the interface debonded energy of ζd/ζm=0.02–0.06 are illustrated in Fig. 10(b), in which the interface debonded length ld/rf decreases with increasing interface debonded length at the same fiber volume content. The broken fibers fraction versus fiber volume fraction curves corresponding to the interface debonded energy of ζd/ζm=0.02–0.06 are illustrated in Fig. 10(c), in which the broken fiber fraction increases with increasing interface debonded energy at the same fiber volume fraction. For C/borosilicate composite, the predicted matrix cracking stress with the interface debonded energy ramge of ζd/ζm=0.005–0.015 agrees with the experimental data corresponding to the fiber volume fraction changing from 30% to 55%, as shown in Fig. 11(a). The interface debonded length ld/rf versus the fiber volume fraction curves corresponding to the interface debonded energy of ζd/ζm=0.005–0.015 are illustrated in Fig. 11(b), in which the interface debonded length ld/ rf decreases with increasing interface debonded length at the same fiber volume content. The broken fibers fraction versus fiber volume fraction curves corresponding to the interface debonded energy of ζd/ ζm=0.005–0.015 are illustrated in Fig. 11(c), in which the broken fiber fraction increases with increasing interface debonded energy at the same fiber volume fraction. From Figs. 9–11, it can be found that the fiber/matrix interface is with strong bonding in SiC/borosilicate composite, and weak bonding in SiC/LAS and C/borosilicate composites.
Acknowledgements The work reported here is supported by the Natural Science Fund of Jiangsu Province (Grant no. BK20140813), and the Fundamental Research Funds for the Central Universities (Grant no. NS2016070). The author would also thank the anonymous reviewers and the editor for their valuable comments on an earlier version of the paper. References [1] R.R. Naslain, Compos. Sci. Technol. 64 (2004) 155–170. [2] J. Aveston, G.A. Cooper, A. Kelly. In: Proceedings of the Conference on The Properties of Fiber Composites, National Physical Laboratory, IPC Science and Technology Press, Guildford, 1971, pp. 15–26. [3] B. Budiansky, J.W. Hutchinson, A.G. Evans, J. Mech. Phys. Solids 34 (1986) 167–189. [4] Y.C. Chiang, Eng. Fract. Mech. 65 (2000) 15–28. [5] D.B. Marshall, B.N. Cox, A.G. Evans, Acta Metall. 33 (1985) 2013–2021. [6] L.N. McCartney, Proc. R. Soc. Lond. 409 (1987) 329–350. [7] J. Aveston, A. Kelly, J. Mater. Sci. 8 (1973) 352–362. [8] V.P. Rajan, F.W. Zok, J. Mech. Phys. Solids 73 (2014) 3–21. [9] M.W. Barsoum, P. Kangutkar, A.S.D. Wang, Compos. Sci. Technol. 44 (1993) 257–269. [10] M.D. Thouless, A.G. Evans, Acta Metall. 36 (1988) 517–522. [11] H.C. Cao, M.D. Thouless, J. Am. Ceram. Soc. 73 (1990) 2091–2094. [12] M. Sutcu, Acta Metall. 37 (1989) 651–661. [13] H.R. Schwietert, P.S. Steif, J. Mech. Phys. Solids 38 (1990) (1990) 325–343. [14] W.A. Curtin, J. Am. Ceram. Soc. 74 (1991) 2837–2845. [15] Y. Weitsman, H. Zhu, J. Mech. Phys. Solids 41 (1993) 351–388. [16] F. Hild, J.M. Domergue, F.A. Leckie, A.G. Evans, Int. J. Solids Struct. 31 (1994) 1035–1045. [17] J.P. Solti, S. Mall, D.D. Robertson, Compos. Sci. Technol. 54 (1995) 55–66. [18] C.D. Cho, J. Mech. Sci. Technol. 10 (1996) 49–56. [19] R. Paar, J.L. Valles, R. Danzer, Mater. Sci. Eng. A 250 (1998) 209–216. [20] K. Liao, R.L. Reifsnider, Int. J. Fract. 106 (2000) 95–115. [21] S.J. Zhou, W.A. Curtin, Acta Metall. Mater. 43 (1995) 3093–3104. [22] R.E. Dutton, N.J. Pagano, R.Y. Kim, T.A. Parthasarathy, J. Am. Ceram. Soc. 83 (2000) 166–174. [23] Z.H. Xia, W.A. Curtin, Acta Mater. 48 (2000) 4879–4892. [24] C.H. Hsueh, Acta Mater. 44 (1996) 2211–2216. [25] Y.C. Gao, Y.W. Mai, B. Cotterell, J. Appl. Math. Phys. 39 (1988) 550–572. [26] Y.J. Sun, R.N. Singh, Acta Mater. 46 (1998) 1657–1667.
8. Conclusions In this paper, the synergistic effects of fiber debonding and fracture on matrix cracking stress of fiber-reinforced CMCs have been investigated using the energy balance approach. The experimental matrix cracking stress of three different CMCs, i.e., SiC/borosilicate, SiC/LAS, and C/borosilicate, with different fiber volume fraction have been predicted. It was found that the fiber/matrix interface possesses strong bonding in SiC/borosilicate composite, and weak bonding in SiC/LAS and C/borosilicate composites. The effects of fiber volume fraction, interface shear stress, interface debonded energy, fiber Weibull modulus, and fiber strength on matrix cracking stress, interface debonded length and fiber broken fraction have been analyzed. (1) When the fiber volume fraction increases, the interface debonded length decreases due to more fibers carrying the applied load,
490