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Micromechanical analysis of fiber and titanium matrix interface by shear lag method
Accepted Manuscript Micromechanical Analysis of Fiber and Titanium Matrix Interface by Shear Lag Method Q. Sun, X. Luo, Y.Q. Yang, B. Huang, N. Jin, W. Zhang, G.M. Zhao PII:
S1359-8368(15)00295-4
DOI:
10.1016/j.compositesb.2015.05.001
Reference:
JCOMB 3588
To appear in:
Composites Part B
Received Date: 30 December 2014 Revised Date:
30 April 2015
Accepted Date: 3 May 2015
Please cite this article as: Sun Q, Luo X, Yang YQ, Huang B, Jin N, Zhang W, Zhao GM, Micromechanical Analysis of Fiber and Titanium Matrix Interface by Shear Lag Method, Composites Part B (2015), doi: 10.1016/j.compositesb.2015.05.001. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Micromechanical Analysis of Fiber and Titanium Matrix Interface by Shear Lag Method Q. Sun, X. Luo*, Y.Q. Yang*, B. Huang, N. Jin, W. Zhang, G.M. Zhao
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State Key Lab of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, P. R. China
Abstract: The model based on fracture mechanics is developed to evaluate the fracture
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toughness Γ of the fiber/matrix interface in titanium alloys reinforced by SiC
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monofilaments. Theoretical model for single fiber push-out testing is obtained by shear-lag method. The influences of several key factors (such as the applied stress needed for crack advance, crack length, and interfacial frictional shear stress) are discussed. Using the model, the interfacial toughness of typical composites including SCS-6/Ti-6-4,
SCS-6/Timetal
834,
SCS-6/Timetal
21s,
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Sigma1240/Ti-6-4,
SCS-6/Ti-24-11 and SCS-6/Ti-15-3 are successfully predicted compared with previous results of these composites. It is verified that the model can reliably predict the
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interfacial toughness of the titanium matrix composites as well as other metal matrix
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composites, due to interfacial debonding usually occurs at the bottom face of the samples in such composites. Keywords:
A. Metal-matrix composites (MMCs); B. Interface/interphase; B. Fracture toughness; Shear lag
*
Corresponding author, E-mail address: [email protected] (XL), [email protected] (YQY) 1
ACCEPTED MANUSCRIPT 1. Introduction SiC fiber reinforced titanium matrix composites (TMCs) have been considered as high temperature structural materials in many applications, such as aerospace and
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motor-mobile industries due to their low density, high performance, high specific strength and stiffness at room and elevated temperatures [1,2]. It is well known that the performance of such composites have been critically influenced by the properties of
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fiber/matrix interface [3−5]. Therefore, determining the interfacial behavior is vital for
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this class of composites. Push-out test, which at first was widely used in ceramic matrix composites (CMCs) [6−8], has been introduced as an important experimental technique owing to the simplicity of preparing a specimen and conducting an experiment. For TMCs, there exists high bonding strength at the interface owing to the strong chemical
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activation of titanium. Thus it is necessary to use thin slices of composites to avoid the fracture of indenter or the crush of fiber [9]. Moreover, the higher thermal residual stresses are induced at fiber/matrix interface owing to the mismatch of thermal
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expansion coefficients between fiber and matrix. These two factors (thinner thickness of
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specimen and higher residual stresses) prompt that interface failure initiates from the bottom face of the specimen [9−16]. It is different from CMCs, in which interface failure initiates at the loaded (top) face [17,18]. Therefore, it is necessary to build new theoretical models for TMCs in order to evaluate their interfacial properties. There are two approaches on the theoretical analysis of the interface debonding in the push-out test. One is based upon the stress (including quadratic [6, 12] and maximum [13,19,20] shear stress) criterion, which is that debonding occurs when the interfacial 2
ACCEPTED MANUSCRIPT stress exceeds the interfacial strength. The other is based on fracture mechanics in which the debonded region is considered as an interfacial crack and its propagation is dependent on the energy balance in terms of interfacial fracture toughness (critical
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strain energy release rate) [21−24]. In the latter, most detailed fracture mechanics, such as crack propagation during the push-out testing and energies of the interfacial debonding have been addressed [25]. Therefore, the fracture mechanics approach is
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more attractive in the analysis of push-out test. Extensive works have been carried out
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to analyze the crack growth behavior and interfacial debonded energies in push-out testing through the fracture mechanics approach. For pull-out test, Hutchinson et al. [26] defined interfacial fracture toughness to be the change in strain energy of the system and the work done by the loading system due to crack propagating an unit area, with
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consideration of Poisson effect and frictional sliding stress. The model was based on shear lag theory and neglected the work done against friction. For push-out test, Dollar [27,28] presented interfacial toughness by cohesive zone model method. Kerans and Zhou
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[29−31] defined the interfacial fracture toughness to be the change in strain energy of
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the system due to crack propagation by shear-lag method. Subsequently, Majumdar [32] considered the work done by applied load to the system due to the crack propagation in addition to the strain energy of the system. On this basis, the work against frictional stress is considered by Kalton et al. [33]. The interfacial toughness above was given under the situation of the top face debonding. However, for almost all thin slice specimens of TMCs, interface failure is likely to initiate at the bottom face during push-out testing. The expressions of the interfacial fracture toughness for the situation of 3
ACCEPTED MANUSCRIPT bottom face failure are different from those for the top face failure owing to the different stress distribution in the debonded and bonded regions. In the case of the bottom face debonding, the interfacial toughness was presented by compliance function by Majumdar
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[32]. Yuan [34] also deduced an expression of the interfacial fracture toughness, including the strain energy UP produced by the applied load, the strain energy UR generated by thermal residual stress, and the strain energy UF consumed by interfacial friction stress.
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This formula also considered such factors as Poisson’s ratio and the effect of the free end
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surface.
In this paper, an analytical model is presented for evaluating the interfacial fracture energy from loading curves obtained during push-out testing. The model is based on shear–lag analysis, taking into account the effects of specimen thickness, fiber volume
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fraction, Poisson’s ratio, interfacial friction coefficient, thermal residual stresses and interfacial frictional sliding stress, and so on. The interface debonding is characterized by a shear fracture (mode 2). Two characterizations of sliding friction are considered:
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Case I, a constant frictional sliding stress τ0, and Case II, a combination of a constant
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frictional sliding stress τ0 and the effect of Poisson contraction of the fiber. Both expressions of Γ are deduced based on Kalton’s basic energy balance equation [33]. In addition, the effects of several key factors are discussed, such as the applied stress needed for crack advance, crack length, and interfacial frictional shear stress. The interfacial
toughness
of
the
composites
Sigma1240/Ti-6-4,
SCS-6/Ti-6-4,
SCS-6/Timetal 834, SCS-6/Timetal 21s, SCS-6/Ti-24-11 and SCS-6/Ti-15-3 is predicted by the two Γ expressions. 4
ACCEPTED MANUSCRIPT 2. Model of micromechanical analysis 2.1. Basic governing equations Two kinds of SiC fiber, C-coated and uncoated, are used for TMCs. The debonding
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occurs between C coating and reaction layer for C-coated SiC fiber in push-out testing, while the debonding occurs between SiC and reaction layer for uncoated SiC fiber [35], as shown in Figure 1. This paper employs a simplified two-phase model neglecting the
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C coating and reaction layer, as C coating (if with C coating) can be considered to be
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together with the fiber, and reaction layer together with the matrix. In the analytical model, it is assumed that the fiber/matrix interface is perfectly bonding before loading, and there is no spontaneous debonding caused by thermal residual stresses. The geometry of the cylinder model is shown in Figure 2, which is widely used to
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evaluate interfacial propertied [25,29−31,36,37]. In the model, a fiber with radius rf is embedded at the centre of a coaxial cylindrical shell of the matrix with a radius rm and a
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total length L. A set of cylindrical coordinates (r, θ, z) is employed, where the z-axis corresponds to the axis of the fiber and r is the perpendicular distance to the z-axis. It is
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assumed that the model of deformation is symmetric about the fiber axis (i.e. axisymmetric) and thus the stress components (σrr, σθθ, σzz, τrz) and the displacement components (ur, uz) are independent of the tangential coordinate θ, and the remaining stress and displacement components are all zero. At the same time, the compliance of the fiber and matrix is neglected. At the end of the specimen, z = 0, the fiber is loaded by a force P, and at the other end, z = L, the matrix is fixed. Therefore, for perfectly
5
ACCEPTED MANUSCRIPT elastic and isotropic fiber and matrix, the general relationships between strains and stresses are 1 {σ zzf (r , z ) - υ f [σ rrf (r , z ) + σ θθf (r , z )]} Ef
for the fiber (i.e. 0 ≤ r ≤ rf), and 1 {σ zzm (r , z ) - υ m [σ rrm (r , z ) + σ θθm (r , z )]} Em
ε rzm (r , z ) =
1 ∂u zm (r , z ) 1 + υ m = τ rzm (r , z ) 2 ∂r Em
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ε zm ( r , z ) =
(1)
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ε zf ( r , z ) =
(2)
(3)
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for the matrix (i.e. rf ≤ r ≤ rm, τ rzm (rf , z ) = τ ( z ) , τ rzm (rm , z ) = 0 , u zm (rf , z ) = u zf (rf , z ) ), where E denotes elastic modulus, υ denotes Poisson's ratio, the subscripts f and m denote fiber and matrix, respectively, and τ(z) represents the interfacial frictional shear stress τd(z) in the debonded region and the interfacial shear stress τb(z) in the bonded
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region. For the matrix shear strain in equation (3), compared to the axial displacement gradient along the r-direction, the radial displacement gradient along the z-direction is
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neglected. In the r-direction, the axial stresses in the fiber and the matrix are assumed as the average stresses to simplify analysis [38], i.e.
σ zzf ( z ) =
2 rf2
σ zzm ( z ) =
2Vf rf2
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∫
rf
0
σ zzf (r , z )rdr
∫
rm
rf
(4)
σ zzm (r , z )rdr
(5)
where Vf (= rf2 /(rm2 − rf2 ) ) is the volume ratio of the fiber to the matrix. The internal stress is transferred from the fiber to the surrounding matrix through the interfacial shear stress. The mechanical equilibrium conditions between fiber, matrix and interface are 6
ACCEPTED MANUSCRIPT σ P = σ zzf ( z ) +
1 σ zzm ( z ) Vf
(6)
dσ zzf ( z ) 2 = − τ ( z) dz rf
(7)
dσ zzm ( z ) 2Vf = τ ( z) dz rf
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r
(8)
∂σ zzm ( z ) ∂rτ rzm (r , z ) + =0 ∂z ∂r
(9)
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2.2. The axial stresses in bonded and debonded region
An interfacial crack is assumed to propagate from the bottom face z = L towards the
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top face z = 0, in the opposite direction of loading. As shown in Figure 3, the specimen is divided into three different regions, i.e. a debonded region I (L ≤ z ≤ L-l), a check tip region II (L-l ≤ z ≤ l1) and a continuous region III (l1 ≤ z ≤ 0).
given as follows.
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For interfacial frictional sliding stress τd(z), two characterizations are considered and
2.2.1. Case I ─ a constant frictional sliding stress, τd(z) = τ0
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The debonded region I locates at the bottom of the specimen. In this region, the interfacial frictional sliding stress is a constant shear stress, τ0. The axial stress in the
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fiber is induced only by the frictional shear stress τ0 since the interface has debonded . The axial stress in the fiber are obtained by integrating equation (7) from L to z, and are given by
σ zzfd ( z ) =
2τ 0 ( L − z ) rf
(10)
which was used to describe the fiber axial stress in the debonded (embedded) region for the situation of the embedded end debonding in pull-out test by Leung [37]. 7
ACCEPTED MANUSCRIPT Hence, at the point L-l, the axial stresses in the fiber are
σ L −l =
2τ 0l rf
(11)
In the region II, the stress distribution is not well-defined because the region is very
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small [33].
Next, in continuous region III (l1 ≤ z ≤ 0), it is assumed that the interface bonds
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perfectly and l1 ≈ L - l. Therefore, the axial stress in the fiber caused by applied load P is obtained by solving the above equations (1-9) with given boundary conditions in this
σ zzf ( z) = −
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region, as follows
A2 1 A A σ+ [( 2 σ + σ L−l ) sinh( A1 z) − ( 2 σ + σ ) sinh( A1 ( z + l − L))] A1 A1 sinh( A1 (L − l )) A1
(12)
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whose solving procedure is detailed in Appendix A.
Since there is axial thermal stress in region III (bonded), all the axial stresses in the fiber is,
σ zzfb (z) = σ zzf (z) − σ z,∆T
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2[α (1 − 2kυ f ) + Vf (1 − 2kυ m )] − 2Vf (1 − 2kυ m ) , A2 = . 2 2 (1 + υ m )[2Vf rm ln(rm / rf ) − rf ] (1 + υ m )[2Vf rm2 ln(rm / rf ) − rf2 ]
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where A1 =
(13)
2.2.2 Case II ─ effect of Poisson expansion In this case, the interfacial frictional sliding stress is characterized by a combination of τ0 and a variation
related to the effect of Poisson contraction of the fiber, µA, which
was used to characterize the interfacial sliding by Kuntz and Liang [17,39], namely
τd(z) = τ0 + µA
A=
αυ f σ zzf ( z ) − υ mσ zzm ( z ) α (1 − υ f ) + 1 + υ m + 2Vf 8
(14)
ACCEPTED MANUSCRIPT where A ( = σ rrf (rf , z ) = σ rrm (rf , z ) ) is the radial stress at the interface caused by Poisson contraction of the fiber and α = Em/Ef [30]. Gao et al. [38] previously obtained the expression of A by considering a thin fiber (i.e. σ rrf ( z ) = σ θθf ( z ) ) and a plane strain
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deformation of the matrix with a stress free cylindrical surface in the radial direction (i.e. σ θθm (rm , z ) = −2Vf σ rrm (rf , z ) , σ rrm (rm , z ) = 0 ). Zhou [30,31,36] has successfully used it to analyze pull-out test and push-out (top face debonding) test. In debonded
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region, the fiber protrudes out from matrix along the axial direction and becomes
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slender because the residual compressive stresses are relieved and the fiber is free at the bottom end [33,40−43]. Thus the influence of Poisson effect to the axial stress in the fiber is produced and is contracting in this region, as shown in equation (14). However, the effect of Poisson’s ratio may be very little since the axial stress in the fiber is low on
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the bottom.
In the debonded region I, the axial stress in the fiber is obtained by substituting equation (14) into equation (7), then integrating equation (7) from L to z, as follows:
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σ zzfd ( z )* = (ωσ − σ 1 )[1 − exp(λ ( L − z ))]
(15)
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where ω = αυ f /(αυ f + Vf υ m ) , σ 1 = τ 0 / µk , k = (αυ f + γυ m ) /[α (1 − υ f ) + 1 + υ m + 2Vf ] and λ = 2µk / rf .
Hence, at the point L-l, the axial stresses in the fiber are
σ L −l * = (ωσ − σ 1 )[1 − exp(λ l )]
(16)
Similarly to the solution of Case I, the axial stress in the fiber is obtained, which as caused by applied load P in the bonded region III, namely
σ zzf (z)* = −
A2 1 A A σ+ [( 2 σ + σ L* −l ) sinh( A1 (L − z)) − ( 2 σ + σ ) sinh( A1 (l − z))] A1 A1 sinh( A1 (L − l)) A1 9
ACCEPTED MANUSCRIPT (17) Then, the all axial stresses in the fiber in bonded region is
σ zzfb (z)* = σ zzf ( z)* − σ z ,∆T
(18)
In this paper, the basic energy balance equation used is
(19)
dU ex dU se dU fr − − dA dA dA
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Γ=
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2.3. The expression of interfacial fracture toughness
where Γ is the crack driving force (interfacial fracture toughness), dUex, dUse and dUfr is
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the work done on the system by the applied load P, the increase in a stain energy of the system and the work against friction, due to a crack increment dA, respectively. In order to obtain interfacial fracture toughness, expressions for dUex, dUse and dUfr are required.
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For Case I, the expressions for dUex, dUse and dUfr are solved by equations (10), (11) and (13), whose solution procedures are detailed in Appendix B, as equations (B3), (B5)
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and (B10). Then the interfacial fracture toughness is obtained by substituting equations (B3),(B5) and (B10) into equation (19), as follows: rf A A [ B1σ P,2 c + C1σ P,cσ 0 + D1σ P,c ( 2 σ P,c − σ z ,∆T ) + E1 ( 2 σ P,c − σ z , ∆T )σ 0 4 Ef A1 A1
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Γ1 =
(20)
A F1σ − ( 2 σ P,c − σ z , ∆T ) 2 − 2σ 0σ z , ∆T l ] A1 2 0
B1 =
where σ0= 2τ0/rf,
2 coshϕ + ϕ cothϕ − 1 A 2 2(1 − coshϕ )(1 + ϕ cothϕ ) + ϕ sinhϕ + ( )[ ] sinh2 ϕ A1 sinh2 ϕ , −(
C1 =
A2 2 2(1 − ϕ cothϕ )(coshϕ − 1) + ϕ sinhϕ )[ ] A1 sinh2 ϕ
ϕ coshϕ + sinhϕ − 2 coshϕ sinhϕ A2 (1 − coshϕ )(sinhϕ + ϕ ) −( ) A1 A1 sinh2 ϕ A1 sinh2 ϕ 2 cosh2 ϕ − 2ϕ cothϕ coshϕ + ϕ sinhϕ A 2(1 − ϕ cothϕ )(coshϕ − 1) + ϕ sinhϕ − l + ( 2 )[ ]l 2 sinh ϕ A1 sinh2 ϕ 10
,
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2(1 − coshϕ ) 2A2 l 1 ( − coshϕ ) , E1 = 2(coshϕ − 1)[ − ], 2 2 sinh ϕ A1 sinh ϕ A1 sinhϕ
1 − ϕ cothϕ 2 coshϕ sinhϕ − ϕ and ϕ = l + l sinh2 ϕ A1 sinh2 ϕ
F1 =
A1 ( L − l ) .
For Case II, an analogous procedure is now carried out. The interfacial fracture
Γ2 =
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toughness Γ2 is obtained by using equations (15), (16) and (18), namely
rf A A [ B2σ P,2 c + C2σ P, c (ωσ P, c − σ 1 ) + D2σ P, c ( 2 σ P, c − σ z , ∆T ) + E2 ( 2 σ P, c − σ z , ∆T ) ⋅ 4 Ef A1 A1
(21)
where
2 coshϕ − ϕ cothϕ − 1 A2 2(1 − coshϕ )(1 + ϕ cothϕ ) + ϕ sinhϕ − ( )[ ] sinh2 ϕ A1 sinh2 ϕ , −(
C2 = [
A2 2 2(1 − ϕ cothϕ )(coshϕ − 1) + ϕ sinhϕ )[ ] A1 sinh2 ϕ
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B2 =
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A (ωσ P, c − σ 1 ) + F2 (ωσ P, c − σ 1 ) − ( 2 σ P, c − σ z , ∆T )2 − 2(ωσ P, c − σ 1 )σ z , ∆T (1 − exp(λl ))] A1
2 coshϕ sinhϕ − ϕ coshϕ − sinhϕ A 2 (1 − coshϕ )(sinhϕ + ϕ ) +( ) ]λ exp(λl ) A1 A1 sinh2 ϕ A1 sinh2 ϕ
−[
2 cosh2 ϕ − 2ϕ cothϕ coshϕ + ϕ sinhϕ A 2 2(1 − ϕ cothϕ )(coshϕ − 1) + ϕ sinhϕ +( ) ](1 − exp(λl )) sinh2 ϕ A1 sinh2 ϕ
2(coshϕ − 1) 2A2 coshϕ − 1 coshϕ − 1 ( − coshϕ ) , E2 = 2[ (1 − exp(λl )) + λ exp(λl )] and sinh2 ϕ A1 sinh2 ϕ A1 sinhϕ
F2 =
cosh2 ϕ − ϕ cothϕ coshϕ sinhϕ − ϕ (1 − exp(λl ))2 − (1 − exp(λl ))λ exp(λl ) 2 sinh ϕ A1 sinh2 ϕ .
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D2 =
− 2 exp(λl )(exp(λl ) − λl − 1)
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3. Discussions
In order to predict the accuracy and validity of the model, a set of parameters is listed in Table 1. The materials are the common TMCs. For a given value of Γ1, the critical applied stress, σP,c, necessary for crack growth, can be obtained by equation (20), as shown in Figure 4, where the relationship of σP,c vs
Γ1 is plotted for three crack lengths. Clearly, the σP,c increases with increasing Γ1. And the longer crack length is, the greater the stress becomes. It is attributed to the increased 11
ACCEPTED MANUSCRIPT shielding of the crack tip provided by the frictional resistance to sliding. The relationship of σP,c vs crack length is plotted in Figure 5 in three situations. Obviously, the σP,c needed at the crack tip will become greater when Γ1 or τ0 raises . An
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increase in Γ1 inhibits crack growth, while increased frictional sliding resistance enhances the crack shielding effect. Compared with τ0, σP,c is more sensitive.
The axial thermal residual stress σz,∆T in the fiber is symmetric about the center of
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specimen thickness and points the two end face of specimen , after the specimen was cut.
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Its direction is in accord with the direction of σzzf(z) on the bottom, but it is opposite to that on the top. Theoretically, σzzf(z) decreases with increasing σz,∆T for the situation of crack initiation at the bottom face. Then all the axial stress in the fiber will become smaller and smaller in the upper portion of specimen so that the stress does not drive
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crack to grow. Thus σP,c is inversely proportional to σz,∆T for the smaller σz,∆T, while σP,c increases in proportion to σz,∆T for the greater σz,∆T. This has been done in Figure 6, which shows predicted dependence of the σP,c on the interfacial toughness for four σz,∆T.
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The σP,c decreases with increasing σz,∆T when σz,∆T is less than 200 MPa, while the σP,c
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increases with increasing σz,∆T when σz,∆T is more than 200 MPa. Figure 7 displays a comparison of the σP,c from Cases I and II as a function of the Γ for three interfacial frictional shear stresses: -100 MPa, -300 MPa and -500 MPa. It is shown that the σP,c is higher for the higher τ0 for a similar Γ, in both Cases I and II. Such a rise in σP,c is to compensate for the increased shielding of the crack tip provided by the frictional resistance to sliding. On the other hand, the σP,c for Case II is smaller than that for Case I when τ0 is -100 MPa; the σP,c for Case II is greater than that for Case I before 12
ACCEPTED MANUSCRIPT 20 J m-2 when τ0 increases to -300 MPa, while it is quite contrary after 20 J m-2; and the
σP,c for Case II is greater than that for Case I when τ0 increases to -500 MPa. It is known that τd(z) will decrease due to the effect of Poisson’s contraction in the debonded region,
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accordingly, the applied stress against τd(z) will decrease. Naturally, at the crack tip, the applied stress in the fiber will decrease owing to the effect of Poisson’s contraction as well, hence σP,c will increase to compensate the decreased applied stress at the crack tip.
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Figure 8 shows a comparison of the σP,c from Cases I and II as a function of the Γ for
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three crack lengths 20 µm, 100 µm and 200 µm. A lower σP,c is required to initiate crack for Case II, which is completely in accord with Figure 7. On the other hand, the greater the crack length is, the more significant the effect of Poisson contraction is, as a consequence of the greater reduction in τd(z).
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Figure 9 shows the contributions of µ on the critical applied stress needed for crack growth, predicted from equation (21). The σP,c decreases with increasing µ for a same interfacial toughness. The effect of Poisson’s ratio will become greater owing to the
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increased µ, then reduction in τb(z) will increase, thus σP,c decreases with increasing µ.
4. Predictions and validations 4.1. Referenced experimental data
The model is applied to predict the push-out behavior of several unidirectional TMCs systems, such as Sigma1240/Ti-6-4, SCS-6/Ti-6-4, SCS-6/Timetal 834, SCS-6/Timetal 21s, SCS-6/Ti-24-ll and SCS-6/Ti-15-3. The thermo-elastic parameters of the fibers and titanium alloys [33, 41, 44−46] are listed in Table 2. The parametric values of the 13
ACCEPTED MANUSCRIPT specimens and the experimental data [21, 33, 41, 46, 47] obtained from load/displacement of push-out testing are shown in Table 3. In this section, it is assumed that l and µ is 20 µm and 0.3, respectively. The σP,c, in the fiber and τ0 can be evaluated
Pmax 2τ Pmax L = rf πrf2
τ 0 = 2πrf
dP f dz
(22)
dτ f dz P
=L
(23)
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σ P, c =
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by using the experimental data and the following equations
debonding fully, τ Pmax and τ
Pf
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where Pmax and Pf is the maximal load and the load against friction after interface is the interfacial shear stress induced by Pmax and Pf,
respectively. The calculated values of σP,c and τ0 for the above several materials are listed in Table 3.
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4.2. Thermal residual stresses analysis
The thermal residual stresses are obtained by finite element method (ABAQUS). The analysis employs a fiber-matrix 2-D axisymmetic model. In this model, the fiber and
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matrix are described with 4-nodes axisymmetic elements while the interface defined
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with 4-nodes cohesive elements generated by duplicating nodes at the interface on fiber and matrix sides. The interphase element size in r-direction is 0 in thick. The stress-free temperature of the Titanium alloy Ti-6-4 [33], Timetal 834 [41], Timetal 21s [21], Ti-24-11 [46] and Ti-15-3 [46] are 800 °C, 770 °C, 875 °C, 820 °C and 815 °C, respectively. The simulated results are listed in Table 4. 4.3. Calculated results
Then the simulated results, the parameters of specimens and experimental data are 14
ACCEPTED MANUSCRIPT substituted into equations (20) and (21), and the calculated results are listed in Table 4. As expected, the Γ1 (or Γ2) values of the composites SCS-6/Ti-6-4, SCS-6/Timetal 834 and SCS-6/Timetal 21s are in good agreement with the previous simulated data. The
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interfacial fracture toughness of the composite SCS-6/Ti-24-11 increases initially as the temperature rises, but then decreases at above 300 °C. The interfacial fracture toughness of the composite SCS-6/Ti-15-3 is 43.4 (or 46.1) at 400 °C, and drops at above 400 °C.
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Majumdar [32] analyzed experimental push-out load data for SCS-6/Ti-15-3 and
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deduced values for Γ of the order of 30-50 J m-2 at room temperature. It is indicated that the present result is in good agreement with Majumdar’s result. Additionally, the Γ2 of all the composites are greater than their Γ1 except SCS-6/Timetal 834. This is attributed
5. Conclusions
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to the greater τ0 of SCS-6/Timetal 834.
The model for single fiber push-out test is developed to evaluate the interfacial
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fracture toughness Γ of TMCs (or other MMCs) for bottom face initiation situation. The
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solutions to Γ are obtained by the shear-lag method. They are different from that for the situation of crack initiation at the top face since the axial stress distributions are different in the fiber. Based on the solutions of Γ, the following conclusions can be drawn: First, the critical applied stresses necessary for crack growth increases with the increase of crack length, Γ and τ0; Second, the applied stress is inversely proportional to
σz,∆T for the smaller σz,∆T, then the applied stress is proportional to σz,∆T for the greater σz,∆T. Third, the σP,c for Case II is smaller than that of Case I for the small τ0; the σP,c for 15
ACCEPTED MANUSCRIPT Case II is greater than that of Case I for the greater τ0. Fourth, the σP,c decreases with increased µ for the same interfacial toughness for Case II. In order to validate the predictions of the model, the interfacial fracture toughness of
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the composites Sigma1240/Ti-6-4 (23 ℃, 600 ℃), SCS-6/Ti-6-4 (23 ℃), SCS-6/Timetal 834 (23 ℃, 530 ℃), SCS-6/Timetal 21s (23 ℃), SCS-6/Ti-24-11 (23 ℃, 300 ℃, 400 ℃, 650℃ and 815℃) and SCS-6/Ti-15-3 (400 ℃, 700℃ and 815℃) are
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obtained by applying the Γ1 solutions, and the results are in good agreement with
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previous results. Thus, the model can be used to predict the interfacial toughness of TMCs for the situation where crack initiates at the bottom face of the specimens. Certainly, the model will be also applicable to the push-out test analysis of other metal matrix composites reinforced by fiber monofilaments for crack initiation at the bottom
Acknowledgements
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face.
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The authors would like to acknowledge the financial supports of the 111 project
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(B08040) of China, the Natural Science Foundation of China (51271147, 51201134 and 51201135),
the
Fundamental
Research
Funds
for
the
Central
Universities
(3102014JCQ01023) and the Research Fund of the State Key Laboratory of Solidification Processing (115-QP-2014).
References [1] Thomas MP, Winstone MR. Longitudinal yielding behaviour of SiC-fibre-reinforced titanium-matrix composites. Compos Sci Technol 1999; 59: 297–303. 16
ACCEPTED MANUSCRIPT [2] Yang YQ, Dudek HJ, Kumpfert J. Interfacial reaction and stability of SCS-6 SiC/Ti-25Al-10Nb-3V-1Mo composite. Mater Sci Eng, A 1998; A246: 213–220. [3] Huang YD, Hort N, Dieringa H, Kainer KU, Liu YL. Microstructural investigations of interfaces in short fiber reinforced AlSi12CuMgNi composites. Acta Mater 2005； 53:3913–3923.
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[4] Shaw LL, Karpur P, Matikas TE. Fracture strength and damage progression of the fiber/matrix interfaces in titanium-based MMCs with different interfacial layers. Compos Part B, 1998, 29,331–339. [5] Majumdar BS, Matikas TE, Miracle DB. Experiments and analysis of fiber fragmentation in single and multiple-fiber SiC/Ti-6Al-4V metal matrix composites. Compos Part B 1998; 29:131–145.
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[6] Chandra N, Ghonem H. Interfacial mechanics of push-out tests: theory and experiments. Compos Part A 2001; 32: 575–584.
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[7] Mandell JF, Chen JH, McGarry FJ. A microdebonding test for in situ assessment of fibre/matrix bond strength in composite materials. Int J Adhes Adhes 1980;1:40–44. [8] Geubelle PH, Baylor J. Impact-induced delamination of composites: a 2D simulation. Compos Part B 1998;29, 589–602. [9] Galbraith JM, Rhyne EP, Koss DA, Hellmann JR. The interfacial failure sequence during fiber pushout in metal matrix composites. Scr Mater 1996; 35: 543–549.
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[10] Rhyne EP, Hellmann JR, Galbraith JM, Koss DA. Thin-slice fiber pushout and specimen bending in metallic matrix composite tests. Scr Metall Mater 1995;32: 547–552.
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[11] Chandra N, Ananth CR. Analysis of interfacial behavior in MMCs and IMCs by the use of thin-slice push-out tests. Compos Sci Technol 1995;54:87–100.
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[12] Ananth CR, Chandra N. Numerical modeling of fiber push-out test in metallic and intermetallic matrix composites mechanics of the failure process. J Compos Mater 1995; 29: 1488–1514. [13] Yuan MN, Yang YQ, Ma ZJ. Analysis of interfacial behavior in titanium matrix composites by using the finite element method (SCS-6/Ti55). Scr Mater 2007; 56: 533–536. [14] Mukherjee S, Ananth CR, Chandra N. Evaluation of fracture toughness of MMC interfaces using thin-slice push-out tests. Scr Mater 1997; 36(11): 1333–1338. [15] Koss DA, Hellmann JR, Kallas MN. Fiber pushout and interfacial shear in metal-matrix composites. J Metals 1993; 46(3): 34–37. [16] Ghosn LJ, Eldridge JI, Kantzos P. Analytical modeling of the interfacial stress state during pushout testing of SCS-6/Ti-Based composites. Acta Metall Mater 1994; 42: 3895–3908. 17
ACCEPTED MANUSCRIPT [17] Kuntz M, Schlapschi KH, Meier B, Grathwohl G. Evaluation of interface parameters in push-out and pull-out tests. Compos 1994; 25(7):476−481. [18] Ye J, Kaw AK. Determination of mechanical properties of fiber−matrix interface from pushout test. Theor Appl Fract Mec 1999; 32:15−25.
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[19] Yuan MN, Yang YQ, Huang B, Wu YJ. Effect of interface reaction on interface shear strength of SiC fiber reinforced titanium matrix composites. Rare Met Mater Eng 2009; 38(8):1321−1324. [20] Galbraith JM, Kallas MN, Koss DA, Hellmann JR. Residual stresses and resulting damage within fibers intersecting a free surface. MRS Proc 1992; 273: 119–126.
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[21] Mukherjee S, Ananth CR, Chandra N. Effects of interface chemistry on the fracture properties of titanium matrix composites. Compos Part A 1998; 29:1213−1219.
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[22] Ananth CR, Voleti SR, Chandra N. Effect of fiber fracture and interfacial debonding on the evolution of damage in metal matrix composites. Compos Part A 1998; 29: 1203−1211. [23] Ma Q, Liang LC, Clarke DR and Hutchinson JW. Mechanics of the push-out process from in situ measurement of the stress distribution along embedded sapphire fibers. Acta Metall Mater 1994; 42(10): 3299−3308. [24] Yuan MN, Yang YQ. Evaluation of interfacial fracture toughness of SiC fibre reinforced titanium-matrix composite using pushout test. Mater Sci Technol 2009; 25(5):561−566.
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[25] Pupurs A, Varna J. 3-D FEM Modeling of fiber/matrix interface debonding in UD composites including surface effects. Mater Sci Eng 2012; 31:1−10. [26] Hutchinson JW, Jensen HM. Models of fiber debonding and pullout in brittle composites with friction. Mech Mater 1990; 9:139−163.
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[27] Dollar A, Steif PS, Wang YC, Hui CY. Analyses of the fiber push-out test. Int J Solids Struct 1993; 30(10):1313−1329.
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[28] Dollar A, Steif PS. Cohesive zone approach to interpreting the fiber push-out test. J Am Ceram Soc 1993; 76: 897−903. [29] Kerans RJ. Theoretical analysis of the fiber pullout and pushout tests. J Am Ceram Soc 1991; 74(7): 1585−1596. [30] Zhou LM, Kim JK, Mai YW. Micromechanical characterisation of fibre/matrix interfaces. Compos Sci Technol 1993; 48: 227−236. [31] Zhou LM, Mai YW, Ye L. Analyses of fibre push-out test based on the fracture mechanics approach. Compos Eng 1995; 5(10-11): 1199−1219. [32] Majumdar BS, Miracle DB. Interface measurements and applications in fiber-reinforced MMCs. Key Eng Mater 1996; 116−117: 153−172. [33] Kalton AF, Howard SJ, Janczak-Rusch J, Clyne TW. Measurement of interfacial 18
ACCEPTED MANUSCRIPT fracture energy by single fibre push-out testing and its application to the titanium silicon carbide system. Acta Mater 1998; 46(9): 3175−3189. [34] Yuan MN, Yang YQ, Huang B, Li JK, Chen Y. Evaluation of interface fracture toughness in SiC fiber reinforced titanium matrix composite. Trans Nonferrous Met Soc China 2008; 18: 925−929.
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[35] Fu YC, Shi NL, Zhang DZ, Yang R. Effect of C coating on the interfacial microstructure and properties of SiC fiber-reinforced Ti matrix composites. Mater Sci Eng A 2006; 426 : 278–282. [36] Zhou LM, Kim JK, Mai YW. Interfacial debonding and fibre pull-out stresses. J Mater Sci 1992; 27: 3155−3166.
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[37] Leung CKY, Li VC. New strength-based model for the debonding of discontinuous fibres in an elastic matrix. J Mater Sci 1991; 26:5996−6010.
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[38] Gao YC, Mai YW, Cotterell B. Fracture of fiber-reinforced materials. J Appl Math Phys 1988; 39: 550−572. [39] Liang C, Hutchinson JW. Mechanics of the fiber pushout test. Mech Mater 1993; 14: 207−221. [40] Kalton AF, Ward-close CM, Clyne TW. Development of the tensioned push-out test for study of fibre/matrix interfaces. Compos 1994; 25(7): 637−644.
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[41] Zeng WD, Peters PWM, Tanaka Y. Interfacial bond strength and fracture energy at room and elevated temperature in titanium matrix composites (SCS-6/Timetal 834). Compos Part A 2002; 33: 1159−1170.
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[42] Mukherjee S, Ananth CR, Chandra N. Effect of residual stresses on the interfacial fracture behavior of metal-matrix composites. Compos Sci Technol 1997; 57: 1501−1512. [43] Chandra N. Evaluation of interfacial fracture toughness using cohesive zone model. Compos Part A 2002; 33: 1433−1447.
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[44] Zheng D, Ghonem H. Fatigue crack growth of SM-1240/timetal-21s metal matrix composites at elevated temperatures. Metall Mater Trans A 1995;26(9): 2469− 2478. [45] Nyakana SL, Fanning JC, Boyer RB. Quik reference guide for β Titanium alloys in the 00s. J Mater Eng Perform 2005; 14(6): 799−811. [46] Eldridge JI, Ebihara BT. Fiber push-out testing apparatus for elevated temperatures. J Mater Res 1994; 9: 1035−1042. [47] Roman I, Jero PD. Interfacial shear behavior of two titanium-based SCS-6 model composites. MRS Proc 1992; 273: 337−342.
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Appendix A The relations between the shear stress in the matrix, τrzm(r,z), and the interfacial shear
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stress, τb(z), is obtained by combining equations (8) and (9) for the boundary condition
τ rzm (rm , z ) = 0 and τ rzm (rf , z ) = τ b ( z ) , as follows: τ rzm (r , z ) =
Vf (rf2 − r 2 ) τ b ( z) rf r
(A.1)
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Combining Equations (3) and (A1) for the boundary condition of axial displacement continuity at the bonded interface (i.e. u zzm (rf , z ) = u zzf (rf , z ) ) and differentiation with
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respect to z gives
dτ b ( z ) r E [ε (r , z ) − ε zf (rf , z )] = f m zm m 2 dz (1 + υm )[2Vf rm ln(rm / rf ) − rf2 ]
(A.2)
Substituting Equations (1,2,7) into (A2), a bivalent differential equation for σ zzf ( z ) is
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obtained under the condition of σ zzf ( z ) = σ θθ f ( z ) , σ θθ m (rm , z ) = −2Vf σ rrm (rf , z ) ,
σ rrm (rm , z ) = 0 (in Ref.[38]), as following:
d 2σ zzf ( z ) − A1σ zzf ( z ) = A 2σ P,c dz 2
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(A.3)
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Then, the expression for the applied axial stress σ zzf ( z ) is obtained by solving Equation (A.3).
Appendix B
The displacement of the loading point is given as following
δ = δ1 + δ 2 + δ 3 = ∫
L −l L
σ zzfd ( z ) Ef
0
σ zzfb ( z )
l1
Ef
dz + 0 + ∫
dz
(B.1)
Subsittuting Equations (10−14) into Equation (B.1), then differentiation dδ is obtained, here l1≈L-l, 20
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2τ l 2τ 1 − cosh A1 ( L − l ) 1 A [ − 0 − ( 2 σ − σ z , ∆T ) + ( 0 ) Ef rf A1 rf A1 sinh A1 ( L − l )
+(
cosh A1 ( L − l ) − 1 2τ l 2A 2 σ + 0 +σ ) ]dl A1 rf sinh 2 A1 ( L − l )
(B.2)
Pmax = σ P,cπ rf2 , the work done by the loading system is
π rf2 Ef
σ P,c [ −
2τ 0 l 2τ 1 − cosh A1 ( L − l ) A − ( 2 σ − σ z , ∆T ) + ( 0 ) rf A1 rf A1 sinh A1 ( L − l )
cosh A1 ( L − l ) − 1 2τ l 2A + ( 2 σ + 0 +σ ) ]dl A1 rf sinh 2 A1 ( L − l )
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dU ex = Pmax d δ =
(B.3)
elastic strain energy density, 1 ε zm ( z ) ， over the volume of the 2
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1 σ zzm ( z )ε zm ( z ) 2
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The strain energy is obtained by integrating the elastic strain energy density
sample. U se = ∫
L −l
L
01 1 ε zfd ( z )σ zzfd ( z ) dV + 0 + ∫ ε zfb ( z )σ zzfb ( z ) dV l 1 2 2
Then, we obtained,
+ [( −(
4 cosh2 ϕ − 2 coshϕ − 2ϕ cothϕ coshϕ + ϕ sinhϕ A 2 2(1 − ϕ cothϕ )(coshϕ − 1) + ϕ sinhϕ −( ) )l sinh2 ϕ A1 sinh2 ϕ
ϕ coshϕ − sinhϕ A2 (1 − coshϕ )(ϕ + sinhϕ ) 2A 2(coshϕ − 1) σ P,c − ( )[ )]σ P,cσ 0 − ( 2 + 1) 2 2 A1 A1 sinh2 ϕ A1 sinh ϕ A1 sinh ϕ
A2 coshϕ − 1 (coshϕ − 1)l A 1 − ϕ cothϕ 2 σ P,c − σ z ,ΔT ) + 2[ − ]σ 0 ( 2 σ P,c − σ z ,ΔT ) − ( l A1 sinh2 ϕ A1 sinh2 ϕ A1 sinhϕ coshϕ sinhϕ − ϕ l )σ 02 2 A1 sinh ϕ
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+
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(
A 2 2 2(1 − ϕ cothϕ )(coshϕ − 1) + ϕ sinhϕ A 2 (ϕ cothϕ − 1)(coshϕ − 1) ϕ cothϕ − 1 2 ) +( ) + ]σ P,c A1 sinh2 ϕ A1 sinh2 ϕ sinh2 ϕ
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dU se = [(
(B.4)
(B.5)
The term Ufr is
U fr = ∫
Area
τ 0 ⋅ υ ( z ) dA
(B.6)
z
σ L − l + σ z , ∆T
L −l
Ef
υ ( z) = ∫ (ε final - ε initial )dz , ε final =
=
σ ( z) 1 2τ 0 ( L − z ) 1 2τ 0 l = ( + σ z ,∆T ) , ε initial = zzfd Ef Ef rf Ef rf
It is noted that the (B.7) is valid when the equation (B.8) is satisfied
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(B.7)
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(B.8)
The Ufr is obtained by substituting equation (B.7) into (B.6) , as follows 2πrf 1 1 1 τ 0 ( Ll 2 − L2l − ( L − l )3 + L3 ) + σ z , ∆T l 2 Ef 3 3 2
and therefore the work done against friction is solved, πrf2 2 Ef
σ 02l 2 + σ 0σ z , ∆T l dl
(B.10)
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dU fr =
(B.9)
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U fr =
22
ACCEPTED MANUSCRIPT Figure Captions Figure 1. SEM micrographs of SiC fibers after push-out test for SiC(C)/Ti–6Al–4V (a and b) and SiC/Ti–6Al–4V (c and d) composites. (a) and (c) show the top face
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whereas (b) and (d) show the bottom face [35]. Figure 2. Idealized fiber push-out model.
Figure 3. Schematic representation of the axial stress in the fiber during push-out test.
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Figure 4. Critical applied stress necessary for crack growth, plotted as a function of
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interfacial toughness, for three crack lengths. Other variables are given in Table 1. Figure 5. Predicted dependence of the applied stress necessary for crack advance on the crack length. Other variables are given in Table 1.
Figure 6. Predicted dependence of the applied stress necessary for crack advance on the
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interfacial toughness. Plots are shown for the four thermal axial axial stresses. Other variables are given in the Table 1. Figure 7. Comparison of the critical applied stresses from Cases I and II as a function of
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the Γ for three frictional shear stresses. Other variables are given in Table 1.
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Figure 8. Comparison of the critical applied stresses from Cases I and II as a function of the Γ for three crack lengths. Other variables are given in Table 1.
Figure 9. The contributions of µ for the critical applied stress as a function of the Γ2 for the crack length 100 µm. Other variables are given in Table 1.
23
ACCEPTED MANUSCRIPT Table 1. Parameters used to examine predictions of the analytical model Ef (GPa) 469 σP,c(MPa) -1300
Em(GPa) 110 σz,∆T(MPa) -500
υf 0.17 σr,∆T(MPa) -300
υm 0.26 τ0(MPa) -100
rf 71 v 0.35
rm 93 µ 0.3
L(µm) 500 Γ(Jm-2) 30
E(GPa)
υ
a (×10-6)
SCS-6 Sigma1240 Ti-6-4 Timetal 834 Timetal 21 Ti-24-11 Ti-15-3
469 400 115-30 115-70 98.2-40 110-43 88.3-44
0.17 0.21 0.36 0.3 0.35 0.26 0.29
4.0 4.0 10-7.3 11.24 9.5-6.8 11-9.0 11-9.7
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Reference [41] [33] [33] [41] [44,45] [46] [46]
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Material System
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Table 2．Thermo-elastic parameters of SiC and titanium alloy
l(µm) 20
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Table 3. The parameters of specimens and data obtained during the push-out test σP,c τ0 (Experimental) (Experimental) Material T rf L v [Reference] [Reference] System (℃) µm (µm) (µm) N MPa Sigma1240/Ti-6-4a 23 50 200 0.33 -1030 -127 Sigma1240/Ti-6-4a 600 50 200 0.33 -466 -28.7 SCS-6/Ti-6-4b 23 71 550 0.35 -1768 -448.6 SCS-6/Timetal 834c 23 71 400 0.4 -1958.5 -717.3 c SCS-6/Timetal 834 530 71 400 0.4 -587.5 -141 SCS-6/Timetal 21sd 23 70 530 0.35 -1559.9 -379.7 23 71 350 0.35 -1074.6 -198.9 300 71 350 0.35 -1123.9 -186.7 SCS-6/Ti-24-11e 400 71 350 0.35 -880.4 -128.3 650 71 350 0.35 -700 -93.3 815 71 350 0.35 -394.4 -21 400 71 350 0.35 -1212.7 -133 SCS-6/Ti-15-3e 700 71 350 0.35 -867.6 -105 815 71 350 0.35 -670.4 -35 The results obtained from: aRef. [33]; bRef. [47]; cRef. [41]; dRef. [21] and eRef. [46]
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Table 4. The thermal residual stresses and the interfacial toughness simulated by other workers and calculated by the equations (16i) and (16ii) of the present model. Γ2 Γ Γ1 Material σz,∆T σr,∆T (Equ.16i) (Equ.16ii) (Simulation) T(℃) -2 (MP) (MP) System (J m ) (J m-2) (J m-2) Sigma1240/Ti-6-4 23 -318a -255a 22.5 25 a Sigma1240/Ti-6-4 600 -21 -24.4a 5.7 6.3 SCS-6/Ti-6-4 23 -815.2 -346 55.3 58.1 52.5b SCS-6/Timetal 834 23 -839 -373 42.2 27.5 40b SCS-6/Timetal 834 530 -175 -73 6.4 6.2 5b SCS-6/Timetal 21s 23 -736 -348 42.6 44.8 50-70c 23 -563 -395 23.3 24.2 300 -312.5 -216 32 33.4 SCS-6/Ti-24-11 400 -134 -112 21 22 650 -58 -34 13.6 14.4 815 0 0 5.2 5.5 400 -237 -161 43.4 46.1 SCS-6/Ti-15-3 700 -63 -52 21.5 22.9 815 0 0 15.1 16.1 The results obtained from: aRef. [33]; bRef. [41]; cRef. [21]
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Figure1. SEM micrographs of SiC fibers after being pushed-out for SiC(C)/ Ti–6Al–4V (a and b) and SiC/Ti–6Al–4V (c and d) composites. (a) and (c)
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show the top face whereas (b) and (d) show the bottom face [35].
Figure 2. Idealized fiber push-out model.
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Figure 3. Schematic representation of the axial stress in the fiber during push-out
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testing.
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Figure 4. Critical applied stress necessary for crack growth, plotted as a function of interfacial toughness, for three crack lengths. Other variables are given in Table 1.
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Figure 5. Predicted dependence of the applied stress necessary for crack advance on
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the crack length. Other variables are given in Table 1.
Figure 6. Predicted dependence of the applied stress necessary for crack advance on
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the interfacial toughness. Plots are shown for the four thermal axial axial stresses. Other variables are given in the Table 1.
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Figure 7. Comparison of the critical applied stresses from Case I and Case II as a
function of the Γ for three frictional shear stresses. Other variables are given in Table
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1.
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Figure 8. Comparison of the critical applied stresses from Case I and Case II as a function of the Γ for three crack lengths. Other variables are given in Table 1.
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Figure 9. The contributions of µ for the critical applied stress as a function of the Γ2 for the crack length 100 µm. Other variables are given in Table 1.
S1359-8368(15)00295-4
DOI:
10.1016/j.compositesb.2015.05.001
Reference:
JCOMB 3588
To appear in:
Composites Part B
Received Date: 30 December 2014 Revised Date:
30 April 2015
Accepted Date: 3 May 2015
Please cite this article as: Sun Q, Luo X, Yang YQ, Huang B, Jin N, Zhang W, Zhao GM, Micromechanical Analysis of Fiber and Titanium Matrix Interface by Shear Lag Method, Composites Part B (2015), doi: 10.1016/j.compositesb.2015.05.001. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Micromechanical Analysis of Fiber and Titanium Matrix Interface by Shear Lag Method Q. Sun, X. Luo*, Y.Q. Yang*, B. Huang, N. Jin, W. Zhang, G.M. Zhao
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State Key Lab of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, P. R. China
Abstract: The model based on fracture mechanics is developed to evaluate the fracture
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toughness Γ of the fiber/matrix interface in titanium alloys reinforced by SiC
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monofilaments. Theoretical model for single fiber push-out testing is obtained by shear-lag method. The influences of several key factors (such as the applied stress needed for crack advance, crack length, and interfacial frictional shear stress) are discussed. Using the model, the interfacial toughness of typical composites including SCS-6/Ti-6-4,
SCS-6/Timetal
834,
SCS-6/Timetal
21s,
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Sigma1240/Ti-6-4,
SCS-6/Ti-24-11 and SCS-6/Ti-15-3 are successfully predicted compared with previous results of these composites. It is verified that the model can reliably predict the
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interfacial toughness of the titanium matrix composites as well as other metal matrix
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composites, due to interfacial debonding usually occurs at the bottom face of the samples in such composites. Keywords:
A. Metal-matrix composites (MMCs); B. Interface/interphase; B. Fracture toughness; Shear lag
*
Corresponding author, E-mail address: [email protected] (XL), [email protected] (YQY) 1
ACCEPTED MANUSCRIPT 1. Introduction SiC fiber reinforced titanium matrix composites (TMCs) have been considered as high temperature structural materials in many applications, such as aerospace and
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motor-mobile industries due to their low density, high performance, high specific strength and stiffness at room and elevated temperatures [1,2]. It is well known that the performance of such composites have been critically influenced by the properties of
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fiber/matrix interface [3−5]. Therefore, determining the interfacial behavior is vital for
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this class of composites. Push-out test, which at first was widely used in ceramic matrix composites (CMCs) [6−8], has been introduced as an important experimental technique owing to the simplicity of preparing a specimen and conducting an experiment. For TMCs, there exists high bonding strength at the interface owing to the strong chemical
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activation of titanium. Thus it is necessary to use thin slices of composites to avoid the fracture of indenter or the crush of fiber [9]. Moreover, the higher thermal residual stresses are induced at fiber/matrix interface owing to the mismatch of thermal
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expansion coefficients between fiber and matrix. These two factors (thinner thickness of
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specimen and higher residual stresses) prompt that interface failure initiates from the bottom face of the specimen [9−16]. It is different from CMCs, in which interface failure initiates at the loaded (top) face [17,18]. Therefore, it is necessary to build new theoretical models for TMCs in order to evaluate their interfacial properties. There are two approaches on the theoretical analysis of the interface debonding in the push-out test. One is based upon the stress (including quadratic [6, 12] and maximum [13,19,20] shear stress) criterion, which is that debonding occurs when the interfacial 2
ACCEPTED MANUSCRIPT stress exceeds the interfacial strength. The other is based on fracture mechanics in which the debonded region is considered as an interfacial crack and its propagation is dependent on the energy balance in terms of interfacial fracture toughness (critical
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strain energy release rate) [21−24]. In the latter, most detailed fracture mechanics, such as crack propagation during the push-out testing and energies of the interfacial debonding have been addressed [25]. Therefore, the fracture mechanics approach is
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more attractive in the analysis of push-out test. Extensive works have been carried out
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to analyze the crack growth behavior and interfacial debonded energies in push-out testing through the fracture mechanics approach. For pull-out test, Hutchinson et al. [26] defined interfacial fracture toughness to be the change in strain energy of the system and the work done by the loading system due to crack propagating an unit area, with
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consideration of Poisson effect and frictional sliding stress. The model was based on shear lag theory and neglected the work done against friction. For push-out test, Dollar [27,28] presented interfacial toughness by cohesive zone model method. Kerans and Zhou
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[29−31] defined the interfacial fracture toughness to be the change in strain energy of
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the system due to crack propagation by shear-lag method. Subsequently, Majumdar [32] considered the work done by applied load to the system due to the crack propagation in addition to the strain energy of the system. On this basis, the work against frictional stress is considered by Kalton et al. [33]. The interfacial toughness above was given under the situation of the top face debonding. However, for almost all thin slice specimens of TMCs, interface failure is likely to initiate at the bottom face during push-out testing. The expressions of the interfacial fracture toughness for the situation of 3
ACCEPTED MANUSCRIPT bottom face failure are different from those for the top face failure owing to the different stress distribution in the debonded and bonded regions. In the case of the bottom face debonding, the interfacial toughness was presented by compliance function by Majumdar
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[32]. Yuan [34] also deduced an expression of the interfacial fracture toughness, including the strain energy UP produced by the applied load, the strain energy UR generated by thermal residual stress, and the strain energy UF consumed by interfacial friction stress.
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This formula also considered such factors as Poisson’s ratio and the effect of the free end
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surface.
In this paper, an analytical model is presented for evaluating the interfacial fracture energy from loading curves obtained during push-out testing. The model is based on shear–lag analysis, taking into account the effects of specimen thickness, fiber volume
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fraction, Poisson’s ratio, interfacial friction coefficient, thermal residual stresses and interfacial frictional sliding stress, and so on. The interface debonding is characterized by a shear fracture (mode 2). Two characterizations of sliding friction are considered:
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Case I, a constant frictional sliding stress τ0, and Case II, a combination of a constant
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frictional sliding stress τ0 and the effect of Poisson contraction of the fiber. Both expressions of Γ are deduced based on Kalton’s basic energy balance equation [33]. In addition, the effects of several key factors are discussed, such as the applied stress needed for crack advance, crack length, and interfacial frictional shear stress. The interfacial
toughness
of
the
composites
Sigma1240/Ti-6-4,
SCS-6/Ti-6-4,
SCS-6/Timetal 834, SCS-6/Timetal 21s, SCS-6/Ti-24-11 and SCS-6/Ti-15-3 is predicted by the two Γ expressions. 4
ACCEPTED MANUSCRIPT 2. Model of micromechanical analysis 2.1. Basic governing equations Two kinds of SiC fiber, C-coated and uncoated, are used for TMCs. The debonding
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occurs between C coating and reaction layer for C-coated SiC fiber in push-out testing, while the debonding occurs between SiC and reaction layer for uncoated SiC fiber [35], as shown in Figure 1. This paper employs a simplified two-phase model neglecting the
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C coating and reaction layer, as C coating (if with C coating) can be considered to be
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together with the fiber, and reaction layer together with the matrix. In the analytical model, it is assumed that the fiber/matrix interface is perfectly bonding before loading, and there is no spontaneous debonding caused by thermal residual stresses. The geometry of the cylinder model is shown in Figure 2, which is widely used to
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evaluate interfacial propertied [25,29−31,36,37]. In the model, a fiber with radius rf is embedded at the centre of a coaxial cylindrical shell of the matrix with a radius rm and a
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total length L. A set of cylindrical coordinates (r, θ, z) is employed, where the z-axis corresponds to the axis of the fiber and r is the perpendicular distance to the z-axis. It is
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assumed that the model of deformation is symmetric about the fiber axis (i.e. axisymmetric) and thus the stress components (σrr, σθθ, σzz, τrz) and the displacement components (ur, uz) are independent of the tangential coordinate θ, and the remaining stress and displacement components are all zero. At the same time, the compliance of the fiber and matrix is neglected. At the end of the specimen, z = 0, the fiber is loaded by a force P, and at the other end, z = L, the matrix is fixed. Therefore, for perfectly
5
ACCEPTED MANUSCRIPT elastic and isotropic fiber and matrix, the general relationships between strains and stresses are 1 {σ zzf (r , z ) - υ f [σ rrf (r , z ) + σ θθf (r , z )]} Ef
for the fiber (i.e. 0 ≤ r ≤ rf), and 1 {σ zzm (r , z ) - υ m [σ rrm (r , z ) + σ θθm (r , z )]} Em
ε rzm (r , z ) =
1 ∂u zm (r , z ) 1 + υ m = τ rzm (r , z ) 2 ∂r Em
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ε zm ( r , z ) =
(1)
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ε zf ( r , z ) =
(2)
(3)
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for the matrix (i.e. rf ≤ r ≤ rm, τ rzm (rf , z ) = τ ( z ) , τ rzm (rm , z ) = 0 , u zm (rf , z ) = u zf (rf , z ) ), where E denotes elastic modulus, υ denotes Poisson's ratio, the subscripts f and m denote fiber and matrix, respectively, and τ(z) represents the interfacial frictional shear stress τd(z) in the debonded region and the interfacial shear stress τb(z) in the bonded
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region. For the matrix shear strain in equation (3), compared to the axial displacement gradient along the r-direction, the radial displacement gradient along the z-direction is
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neglected. In the r-direction, the axial stresses in the fiber and the matrix are assumed as the average stresses to simplify analysis [38], i.e.
σ zzf ( z ) =
2 rf2
σ zzm ( z ) =
2Vf rf2
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∫
rf
0
σ zzf (r , z )rdr
∫
rm
rf
(4)
σ zzm (r , z )rdr
(5)
where Vf (= rf2 /(rm2 − rf2 ) ) is the volume ratio of the fiber to the matrix. The internal stress is transferred from the fiber to the surrounding matrix through the interfacial shear stress. The mechanical equilibrium conditions between fiber, matrix and interface are 6
ACCEPTED MANUSCRIPT σ P = σ zzf ( z ) +
1 σ zzm ( z ) Vf
(6)
dσ zzf ( z ) 2 = − τ ( z) dz rf
(7)
dσ zzm ( z ) 2Vf = τ ( z) dz rf
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r
(8)
∂σ zzm ( z ) ∂rτ rzm (r , z ) + =0 ∂z ∂r
(9)
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2.2. The axial stresses in bonded and debonded region
An interfacial crack is assumed to propagate from the bottom face z = L towards the
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top face z = 0, in the opposite direction of loading. As shown in Figure 3, the specimen is divided into three different regions, i.e. a debonded region I (L ≤ z ≤ L-l), a check tip region II (L-l ≤ z ≤ l1) and a continuous region III (l1 ≤ z ≤ 0).
given as follows.
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For interfacial frictional sliding stress τd(z), two characterizations are considered and
2.2.1. Case I ─ a constant frictional sliding stress, τd(z) = τ0
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The debonded region I locates at the bottom of the specimen. In this region, the interfacial frictional sliding stress is a constant shear stress, τ0. The axial stress in the
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fiber is induced only by the frictional shear stress τ0 since the interface has debonded . The axial stress in the fiber are obtained by integrating equation (7) from L to z, and are given by
σ zzfd ( z ) =
2τ 0 ( L − z ) rf
(10)
which was used to describe the fiber axial stress in the debonded (embedded) region for the situation of the embedded end debonding in pull-out test by Leung [37]. 7
ACCEPTED MANUSCRIPT Hence, at the point L-l, the axial stresses in the fiber are
σ L −l =
2τ 0l rf
(11)
In the region II, the stress distribution is not well-defined because the region is very
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small [33].
Next, in continuous region III (l1 ≤ z ≤ 0), it is assumed that the interface bonds
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perfectly and l1 ≈ L - l. Therefore, the axial stress in the fiber caused by applied load P is obtained by solving the above equations (1-9) with given boundary conditions in this
σ zzf ( z) = −
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region, as follows
A2 1 A A σ+ [( 2 σ + σ L−l ) sinh( A1 z) − ( 2 σ + σ ) sinh( A1 ( z + l − L))] A1 A1 sinh( A1 (L − l )) A1
(12)
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whose solving procedure is detailed in Appendix A.
Since there is axial thermal stress in region III (bonded), all the axial stresses in the fiber is,
σ zzfb (z) = σ zzf (z) − σ z,∆T
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2[α (1 − 2kυ f ) + Vf (1 − 2kυ m )] − 2Vf (1 − 2kυ m ) , A2 = . 2 2 (1 + υ m )[2Vf rm ln(rm / rf ) − rf ] (1 + υ m )[2Vf rm2 ln(rm / rf ) − rf2 ]
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where A1 =
(13)
2.2.2 Case II ─ effect of Poisson expansion In this case, the interfacial frictional sliding stress is characterized by a combination of τ0 and a variation
related to the effect of Poisson contraction of the fiber, µA, which
was used to characterize the interfacial sliding by Kuntz and Liang [17,39], namely
τd(z) = τ0 + µA
A=
αυ f σ zzf ( z ) − υ mσ zzm ( z ) α (1 − υ f ) + 1 + υ m + 2Vf 8
(14)
ACCEPTED MANUSCRIPT where A ( = σ rrf (rf , z ) = σ rrm (rf , z ) ) is the radial stress at the interface caused by Poisson contraction of the fiber and α = Em/Ef [30]. Gao et al. [38] previously obtained the expression of A by considering a thin fiber (i.e. σ rrf ( z ) = σ θθf ( z ) ) and a plane strain
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deformation of the matrix with a stress free cylindrical surface in the radial direction (i.e. σ θθm (rm , z ) = −2Vf σ rrm (rf , z ) , σ rrm (rm , z ) = 0 ). Zhou [30,31,36] has successfully used it to analyze pull-out test and push-out (top face debonding) test. In debonded
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region, the fiber protrudes out from matrix along the axial direction and becomes
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slender because the residual compressive stresses are relieved and the fiber is free at the bottom end [33,40−43]. Thus the influence of Poisson effect to the axial stress in the fiber is produced and is contracting in this region, as shown in equation (14). However, the effect of Poisson’s ratio may be very little since the axial stress in the fiber is low on
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the bottom.
In the debonded region I, the axial stress in the fiber is obtained by substituting equation (14) into equation (7), then integrating equation (7) from L to z, as follows:
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σ zzfd ( z )* = (ωσ − σ 1 )[1 − exp(λ ( L − z ))]
(15)
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where ω = αυ f /(αυ f + Vf υ m ) , σ 1 = τ 0 / µk , k = (αυ f + γυ m ) /[α (1 − υ f ) + 1 + υ m + 2Vf ] and λ = 2µk / rf .
Hence, at the point L-l, the axial stresses in the fiber are
σ L −l * = (ωσ − σ 1 )[1 − exp(λ l )]
(16)
Similarly to the solution of Case I, the axial stress in the fiber is obtained, which as caused by applied load P in the bonded region III, namely
σ zzf (z)* = −
A2 1 A A σ+ [( 2 σ + σ L* −l ) sinh( A1 (L − z)) − ( 2 σ + σ ) sinh( A1 (l − z))] A1 A1 sinh( A1 (L − l)) A1 9
ACCEPTED MANUSCRIPT (17) Then, the all axial stresses in the fiber in bonded region is
σ zzfb (z)* = σ zzf ( z)* − σ z ,∆T
(18)
In this paper, the basic energy balance equation used is
(19)
dU ex dU se dU fr − − dA dA dA
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Γ=
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2.3. The expression of interfacial fracture toughness
where Γ is the crack driving force (interfacial fracture toughness), dUex, dUse and dUfr is
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the work done on the system by the applied load P, the increase in a stain energy of the system and the work against friction, due to a crack increment dA, respectively. In order to obtain interfacial fracture toughness, expressions for dUex, dUse and dUfr are required.
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For Case I, the expressions for dUex, dUse and dUfr are solved by equations (10), (11) and (13), whose solution procedures are detailed in Appendix B, as equations (B3), (B5)
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and (B10). Then the interfacial fracture toughness is obtained by substituting equations (B3),(B5) and (B10) into equation (19), as follows: rf A A [ B1σ P,2 c + C1σ P,cσ 0 + D1σ P,c ( 2 σ P,c − σ z ,∆T ) + E1 ( 2 σ P,c − σ z , ∆T )σ 0 4 Ef A1 A1
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Γ1 =
(20)
A F1σ − ( 2 σ P,c − σ z , ∆T ) 2 − 2σ 0σ z , ∆T l ] A1 2 0
B1 =
where σ0= 2τ0/rf,
2 coshϕ + ϕ cothϕ − 1 A 2 2(1 − coshϕ )(1 + ϕ cothϕ ) + ϕ sinhϕ + ( )[ ] sinh2 ϕ A1 sinh2 ϕ , −(
C1 =
A2 2 2(1 − ϕ cothϕ )(coshϕ − 1) + ϕ sinhϕ )[ ] A1 sinh2 ϕ
ϕ coshϕ + sinhϕ − 2 coshϕ sinhϕ A2 (1 − coshϕ )(sinhϕ + ϕ ) −( ) A1 A1 sinh2 ϕ A1 sinh2 ϕ 2 cosh2 ϕ − 2ϕ cothϕ coshϕ + ϕ sinhϕ A 2(1 − ϕ cothϕ )(coshϕ − 1) + ϕ sinhϕ − l + ( 2 )[ ]l 2 sinh ϕ A1 sinh2 ϕ 10
,
ACCEPTED MANUSCRIPT D1 = −
2(1 − coshϕ ) 2A2 l 1 ( − coshϕ ) , E1 = 2(coshϕ − 1)[ − ], 2 2 sinh ϕ A1 sinh ϕ A1 sinhϕ
1 − ϕ cothϕ 2 coshϕ sinhϕ − ϕ and ϕ = l + l sinh2 ϕ A1 sinh2 ϕ
F1 =
A1 ( L − l ) .
For Case II, an analogous procedure is now carried out. The interfacial fracture
Γ2 =
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toughness Γ2 is obtained by using equations (15), (16) and (18), namely
rf A A [ B2σ P,2 c + C2σ P, c (ωσ P, c − σ 1 ) + D2σ P, c ( 2 σ P, c − σ z , ∆T ) + E2 ( 2 σ P, c − σ z , ∆T ) ⋅ 4 Ef A1 A1
(21)
where
2 coshϕ − ϕ cothϕ − 1 A2 2(1 − coshϕ )(1 + ϕ cothϕ ) + ϕ sinhϕ − ( )[ ] sinh2 ϕ A1 sinh2 ϕ , −(
C2 = [
A2 2 2(1 − ϕ cothϕ )(coshϕ − 1) + ϕ sinhϕ )[ ] A1 sinh2 ϕ
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B2 =
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A (ωσ P, c − σ 1 ) + F2 (ωσ P, c − σ 1 ) − ( 2 σ P, c − σ z , ∆T )2 − 2(ωσ P, c − σ 1 )σ z , ∆T (1 − exp(λl ))] A1
2 coshϕ sinhϕ − ϕ coshϕ − sinhϕ A 2 (1 − coshϕ )(sinhϕ + ϕ ) +( ) ]λ exp(λl ) A1 A1 sinh2 ϕ A1 sinh2 ϕ
−[
2 cosh2 ϕ − 2ϕ cothϕ coshϕ + ϕ sinhϕ A 2 2(1 − ϕ cothϕ )(coshϕ − 1) + ϕ sinhϕ +( ) ](1 − exp(λl )) sinh2 ϕ A1 sinh2 ϕ
2(coshϕ − 1) 2A2 coshϕ − 1 coshϕ − 1 ( − coshϕ ) , E2 = 2[ (1 − exp(λl )) + λ exp(λl )] and sinh2 ϕ A1 sinh2 ϕ A1 sinhϕ
F2 =
cosh2 ϕ − ϕ cothϕ coshϕ sinhϕ − ϕ (1 − exp(λl ))2 − (1 − exp(λl ))λ exp(λl ) 2 sinh ϕ A1 sinh2 ϕ .
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D2 =
− 2 exp(λl )(exp(λl ) − λl − 1)
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3. Discussions
In order to predict the accuracy and validity of the model, a set of parameters is listed in Table 1. The materials are the common TMCs. For a given value of Γ1, the critical applied stress, σP,c, necessary for crack growth, can be obtained by equation (20), as shown in Figure 4, where the relationship of σP,c vs
Γ1 is plotted for three crack lengths. Clearly, the σP,c increases with increasing Γ1. And the longer crack length is, the greater the stress becomes. It is attributed to the increased 11
ACCEPTED MANUSCRIPT shielding of the crack tip provided by the frictional resistance to sliding. The relationship of σP,c vs crack length is plotted in Figure 5 in three situations. Obviously, the σP,c needed at the crack tip will become greater when Γ1 or τ0 raises . An
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increase in Γ1 inhibits crack growth, while increased frictional sliding resistance enhances the crack shielding effect. Compared with τ0, σP,c is more sensitive.
The axial thermal residual stress σz,∆T in the fiber is symmetric about the center of
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specimen thickness and points the two end face of specimen , after the specimen was cut.
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Its direction is in accord with the direction of σzzf(z) on the bottom, but it is opposite to that on the top. Theoretically, σzzf(z) decreases with increasing σz,∆T for the situation of crack initiation at the bottom face. Then all the axial stress in the fiber will become smaller and smaller in the upper portion of specimen so that the stress does not drive
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crack to grow. Thus σP,c is inversely proportional to σz,∆T for the smaller σz,∆T, while σP,c increases in proportion to σz,∆T for the greater σz,∆T. This has been done in Figure 6, which shows predicted dependence of the σP,c on the interfacial toughness for four σz,∆T.
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The σP,c decreases with increasing σz,∆T when σz,∆T is less than 200 MPa, while the σP,c
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increases with increasing σz,∆T when σz,∆T is more than 200 MPa. Figure 7 displays a comparison of the σP,c from Cases I and II as a function of the Γ for three interfacial frictional shear stresses: -100 MPa, -300 MPa and -500 MPa. It is shown that the σP,c is higher for the higher τ0 for a similar Γ, in both Cases I and II. Such a rise in σP,c is to compensate for the increased shielding of the crack tip provided by the frictional resistance to sliding. On the other hand, the σP,c for Case II is smaller than that for Case I when τ0 is -100 MPa; the σP,c for Case II is greater than that for Case I before 12
ACCEPTED MANUSCRIPT 20 J m-2 when τ0 increases to -300 MPa, while it is quite contrary after 20 J m-2; and the
σP,c for Case II is greater than that for Case I when τ0 increases to -500 MPa. It is known that τd(z) will decrease due to the effect of Poisson’s contraction in the debonded region,
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accordingly, the applied stress against τd(z) will decrease. Naturally, at the crack tip, the applied stress in the fiber will decrease owing to the effect of Poisson’s contraction as well, hence σP,c will increase to compensate the decreased applied stress at the crack tip.
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Figure 8 shows a comparison of the σP,c from Cases I and II as a function of the Γ for
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three crack lengths 20 µm, 100 µm and 200 µm. A lower σP,c is required to initiate crack for Case II, which is completely in accord with Figure 7. On the other hand, the greater the crack length is, the more significant the effect of Poisson contraction is, as a consequence of the greater reduction in τd(z).
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Figure 9 shows the contributions of µ on the critical applied stress needed for crack growth, predicted from equation (21). The σP,c decreases with increasing µ for a same interfacial toughness. The effect of Poisson’s ratio will become greater owing to the
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increased µ, then reduction in τb(z) will increase, thus σP,c decreases with increasing µ.
4. Predictions and validations 4.1. Referenced experimental data
The model is applied to predict the push-out behavior of several unidirectional TMCs systems, such as Sigma1240/Ti-6-4, SCS-6/Ti-6-4, SCS-6/Timetal 834, SCS-6/Timetal 21s, SCS-6/Ti-24-ll and SCS-6/Ti-15-3. The thermo-elastic parameters of the fibers and titanium alloys [33, 41, 44−46] are listed in Table 2. The parametric values of the 13
ACCEPTED MANUSCRIPT specimens and the experimental data [21, 33, 41, 46, 47] obtained from load/displacement of push-out testing are shown in Table 3. In this section, it is assumed that l and µ is 20 µm and 0.3, respectively. The σP,c, in the fiber and τ0 can be evaluated
Pmax 2τ Pmax L = rf πrf2
τ 0 = 2πrf
dP f dz
(22)
dτ f dz P
=L
(23)
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σ P, c =
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by using the experimental data and the following equations
debonding fully, τ Pmax and τ
Pf
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where Pmax and Pf is the maximal load and the load against friction after interface is the interfacial shear stress induced by Pmax and Pf,
respectively. The calculated values of σP,c and τ0 for the above several materials are listed in Table 3.
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4.2. Thermal residual stresses analysis
The thermal residual stresses are obtained by finite element method (ABAQUS). The analysis employs a fiber-matrix 2-D axisymmetic model. In this model, the fiber and
EP
matrix are described with 4-nodes axisymmetic elements while the interface defined
AC C
with 4-nodes cohesive elements generated by duplicating nodes at the interface on fiber and matrix sides. The interphase element size in r-direction is 0 in thick. The stress-free temperature of the Titanium alloy Ti-6-4 [33], Timetal 834 [41], Timetal 21s [21], Ti-24-11 [46] and Ti-15-3 [46] are 800 °C, 770 °C, 875 °C, 820 °C and 815 °C, respectively. The simulated results are listed in Table 4. 4.3. Calculated results
Then the simulated results, the parameters of specimens and experimental data are 14
ACCEPTED MANUSCRIPT substituted into equations (20) and (21), and the calculated results are listed in Table 4. As expected, the Γ1 (or Γ2) values of the composites SCS-6/Ti-6-4, SCS-6/Timetal 834 and SCS-6/Timetal 21s are in good agreement with the previous simulated data. The
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interfacial fracture toughness of the composite SCS-6/Ti-24-11 increases initially as the temperature rises, but then decreases at above 300 °C. The interfacial fracture toughness of the composite SCS-6/Ti-15-3 is 43.4 (or 46.1) at 400 °C, and drops at above 400 °C.
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Majumdar [32] analyzed experimental push-out load data for SCS-6/Ti-15-3 and
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deduced values for Γ of the order of 30-50 J m-2 at room temperature. It is indicated that the present result is in good agreement with Majumdar’s result. Additionally, the Γ2 of all the composites are greater than their Γ1 except SCS-6/Timetal 834. This is attributed
5. Conclusions
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to the greater τ0 of SCS-6/Timetal 834.
The model for single fiber push-out test is developed to evaluate the interfacial
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fracture toughness Γ of TMCs (or other MMCs) for bottom face initiation situation. The
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solutions to Γ are obtained by the shear-lag method. They are different from that for the situation of crack initiation at the top face since the axial stress distributions are different in the fiber. Based on the solutions of Γ, the following conclusions can be drawn: First, the critical applied stresses necessary for crack growth increases with the increase of crack length, Γ and τ0; Second, the applied stress is inversely proportional to
σz,∆T for the smaller σz,∆T, then the applied stress is proportional to σz,∆T for the greater σz,∆T. Third, the σP,c for Case II is smaller than that of Case I for the small τ0; the σP,c for 15
ACCEPTED MANUSCRIPT Case II is greater than that of Case I for the greater τ0. Fourth, the σP,c decreases with increased µ for the same interfacial toughness for Case II. In order to validate the predictions of the model, the interfacial fracture toughness of
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the composites Sigma1240/Ti-6-4 (23 ℃, 600 ℃), SCS-6/Ti-6-4 (23 ℃), SCS-6/Timetal 834 (23 ℃, 530 ℃), SCS-6/Timetal 21s (23 ℃), SCS-6/Ti-24-11 (23 ℃, 300 ℃, 400 ℃, 650℃ and 815℃) and SCS-6/Ti-15-3 (400 ℃, 700℃ and 815℃) are
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obtained by applying the Γ1 solutions, and the results are in good agreement with
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previous results. Thus, the model can be used to predict the interfacial toughness of TMCs for the situation where crack initiates at the bottom face of the specimens. Certainly, the model will be also applicable to the push-out test analysis of other metal matrix composites reinforced by fiber monofilaments for crack initiation at the bottom
Acknowledgements
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face.
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The authors would like to acknowledge the financial supports of the 111 project
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(B08040) of China, the Natural Science Foundation of China (51271147, 51201134 and 51201135),
the
Fundamental
Research
Funds
for
the
Central
Universities
(3102014JCQ01023) and the Research Fund of the State Key Laboratory of Solidification Processing (115-QP-2014).
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[4] Shaw LL, Karpur P, Matikas TE. Fracture strength and damage progression of the fiber/matrix interfaces in titanium-based MMCs with different interfacial layers. Compos Part B, 1998, 29,331–339. [5] Majumdar BS, Matikas TE, Miracle DB. Experiments and analysis of fiber fragmentation in single and multiple-fiber SiC/Ti-6Al-4V metal matrix composites. Compos Part B 1998; 29:131–145.
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[6] Chandra N, Ghonem H. Interfacial mechanics of push-out tests: theory and experiments. Compos Part A 2001; 32: 575–584.
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[10] Rhyne EP, Hellmann JR, Galbraith JM, Koss DA. Thin-slice fiber pushout and specimen bending in metallic matrix composite tests. Scr Metall Mater 1995;32: 547–552.
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ACCEPTED MANUSCRIPT [17] Kuntz M, Schlapschi KH, Meier B, Grathwohl G. Evaluation of interface parameters in push-out and pull-out tests. Compos 1994; 25(7):476−481. [18] Ye J, Kaw AK. Determination of mechanical properties of fiber−matrix interface from pushout test. Theor Appl Fract Mec 1999; 32:15−25.
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[19] Yuan MN, Yang YQ, Huang B, Wu YJ. Effect of interface reaction on interface shear strength of SiC fiber reinforced titanium matrix composites. Rare Met Mater Eng 2009; 38(8):1321−1324. [20] Galbraith JM, Kallas MN, Koss DA, Hellmann JR. Residual stresses and resulting damage within fibers intersecting a free surface. MRS Proc 1992; 273: 119–126.
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[21] Mukherjee S, Ananth CR, Chandra N. Effects of interface chemistry on the fracture properties of titanium matrix composites. Compos Part A 1998; 29:1213−1219.
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[22] Ananth CR, Voleti SR, Chandra N. Effect of fiber fracture and interfacial debonding on the evolution of damage in metal matrix composites. Compos Part A 1998; 29: 1203−1211. [23] Ma Q, Liang LC, Clarke DR and Hutchinson JW. Mechanics of the push-out process from in situ measurement of the stress distribution along embedded sapphire fibers. Acta Metall Mater 1994; 42(10): 3299−3308. [24] Yuan MN, Yang YQ. Evaluation of interfacial fracture toughness of SiC fibre reinforced titanium-matrix composite using pushout test. Mater Sci Technol 2009; 25(5):561−566.
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[25] Pupurs A, Varna J. 3-D FEM Modeling of fiber/matrix interface debonding in UD composites including surface effects. Mater Sci Eng 2012; 31:1−10. [26] Hutchinson JW, Jensen HM. Models of fiber debonding and pullout in brittle composites with friction. Mech Mater 1990; 9:139−163.
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[27] Dollar A, Steif PS, Wang YC, Hui CY. Analyses of the fiber push-out test. Int J Solids Struct 1993; 30(10):1313−1329.
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[28] Dollar A, Steif PS. Cohesive zone approach to interpreting the fiber push-out test. J Am Ceram Soc 1993; 76: 897−903. [29] Kerans RJ. Theoretical analysis of the fiber pullout and pushout tests. J Am Ceram Soc 1991; 74(7): 1585−1596. [30] Zhou LM, Kim JK, Mai YW. Micromechanical characterisation of fibre/matrix interfaces. Compos Sci Technol 1993; 48: 227−236. [31] Zhou LM, Mai YW, Ye L. Analyses of fibre push-out test based on the fracture mechanics approach. Compos Eng 1995; 5(10-11): 1199−1219. [32] Majumdar BS, Miracle DB. Interface measurements and applications in fiber-reinforced MMCs. Key Eng Mater 1996; 116−117: 153−172. [33] Kalton AF, Howard SJ, Janczak-Rusch J, Clyne TW. Measurement of interfacial 18
ACCEPTED MANUSCRIPT fracture energy by single fibre push-out testing and its application to the titanium silicon carbide system. Acta Mater 1998; 46(9): 3175−3189. [34] Yuan MN, Yang YQ, Huang B, Li JK, Chen Y. Evaluation of interface fracture toughness in SiC fiber reinforced titanium matrix composite. Trans Nonferrous Met Soc China 2008; 18: 925−929.
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[35] Fu YC, Shi NL, Zhang DZ, Yang R. Effect of C coating on the interfacial microstructure and properties of SiC fiber-reinforced Ti matrix composites. Mater Sci Eng A 2006; 426 : 278–282. [36] Zhou LM, Kim JK, Mai YW. Interfacial debonding and fibre pull-out stresses. J Mater Sci 1992; 27: 3155−3166.
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[37] Leung CKY, Li VC. New strength-based model for the debonding of discontinuous fibres in an elastic matrix. J Mater Sci 1991; 26:5996−6010.
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[38] Gao YC, Mai YW, Cotterell B. Fracture of fiber-reinforced materials. J Appl Math Phys 1988; 39: 550−572. [39] Liang C, Hutchinson JW. Mechanics of the fiber pushout test. Mech Mater 1993; 14: 207−221. [40] Kalton AF, Ward-close CM, Clyne TW. Development of the tensioned push-out test for study of fibre/matrix interfaces. Compos 1994; 25(7): 637−644.
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[41] Zeng WD, Peters PWM, Tanaka Y. Interfacial bond strength and fracture energy at room and elevated temperature in titanium matrix composites (SCS-6/Timetal 834). Compos Part A 2002; 33: 1159−1170.
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[42] Mukherjee S, Ananth CR, Chandra N. Effect of residual stresses on the interfacial fracture behavior of metal-matrix composites. Compos Sci Technol 1997; 57: 1501−1512. [43] Chandra N. Evaluation of interfacial fracture toughness using cohesive zone model. Compos Part A 2002; 33: 1433−1447.
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[44] Zheng D, Ghonem H. Fatigue crack growth of SM-1240/timetal-21s metal matrix composites at elevated temperatures. Metall Mater Trans A 1995;26(9): 2469− 2478. [45] Nyakana SL, Fanning JC, Boyer RB. Quik reference guide for β Titanium alloys in the 00s. J Mater Eng Perform 2005; 14(6): 799−811. [46] Eldridge JI, Ebihara BT. Fiber push-out testing apparatus for elevated temperatures. J Mater Res 1994; 9: 1035−1042. [47] Roman I, Jero PD. Interfacial shear behavior of two titanium-based SCS-6 model composites. MRS Proc 1992; 273: 337−342.
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Appendix A The relations between the shear stress in the matrix, τrzm(r,z), and the interfacial shear
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stress, τb(z), is obtained by combining equations (8) and (9) for the boundary condition
τ rzm (rm , z ) = 0 and τ rzm (rf , z ) = τ b ( z ) , as follows: τ rzm (r , z ) =
Vf (rf2 − r 2 ) τ b ( z) rf r
(A.1)
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Combining Equations (3) and (A1) for the boundary condition of axial displacement continuity at the bonded interface (i.e. u zzm (rf , z ) = u zzf (rf , z ) ) and differentiation with
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respect to z gives
dτ b ( z ) r E [ε (r , z ) − ε zf (rf , z )] = f m zm m 2 dz (1 + υm )[2Vf rm ln(rm / rf ) − rf2 ]
(A.2)
Substituting Equations (1,2,7) into (A2), a bivalent differential equation for σ zzf ( z ) is
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obtained under the condition of σ zzf ( z ) = σ θθ f ( z ) , σ θθ m (rm , z ) = −2Vf σ rrm (rf , z ) ,
σ rrm (rm , z ) = 0 (in Ref.[38]), as following:
d 2σ zzf ( z ) − A1σ zzf ( z ) = A 2σ P,c dz 2
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(A.3)
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Then, the expression for the applied axial stress σ zzf ( z ) is obtained by solving Equation (A.3).
Appendix B
The displacement of the loading point is given as following
δ = δ1 + δ 2 + δ 3 = ∫
L −l L
σ zzfd ( z ) Ef
0
σ zzfb ( z )
l1
Ef
dz + 0 + ∫
dz
(B.1)
Subsittuting Equations (10−14) into Equation (B.1), then differentiation dδ is obtained, here l1≈L-l, 20
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2τ l 2τ 1 − cosh A1 ( L − l ) 1 A [ − 0 − ( 2 σ − σ z , ∆T ) + ( 0 ) Ef rf A1 rf A1 sinh A1 ( L − l )
+(
cosh A1 ( L − l ) − 1 2τ l 2A 2 σ + 0 +σ ) ]dl A1 rf sinh 2 A1 ( L − l )
(B.2)
Pmax = σ P,cπ rf2 , the work done by the loading system is
π rf2 Ef
σ P,c [ −
2τ 0 l 2τ 1 − cosh A1 ( L − l ) A − ( 2 σ − σ z , ∆T ) + ( 0 ) rf A1 rf A1 sinh A1 ( L − l )
cosh A1 ( L − l ) − 1 2τ l 2A + ( 2 σ + 0 +σ ) ]dl A1 rf sinh 2 A1 ( L − l )
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dU ex = Pmax d δ =
(B.3)
elastic strain energy density, 1 ε zm ( z ) ， over the volume of the 2
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1 σ zzm ( z )ε zm ( z ) 2
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The strain energy is obtained by integrating the elastic strain energy density
sample. U se = ∫
L −l
L
01 1 ε zfd ( z )σ zzfd ( z ) dV + 0 + ∫ ε zfb ( z )σ zzfb ( z ) dV l 1 2 2
Then, we obtained,
+ [( −(
4 cosh2 ϕ − 2 coshϕ − 2ϕ cothϕ coshϕ + ϕ sinhϕ A 2 2(1 − ϕ cothϕ )(coshϕ − 1) + ϕ sinhϕ −( ) )l sinh2 ϕ A1 sinh2 ϕ
ϕ coshϕ − sinhϕ A2 (1 − coshϕ )(ϕ + sinhϕ ) 2A 2(coshϕ − 1) σ P,c − ( )[ )]σ P,cσ 0 − ( 2 + 1) 2 2 A1 A1 sinh2 ϕ A1 sinh ϕ A1 sinh ϕ
A2 coshϕ − 1 (coshϕ − 1)l A 1 − ϕ cothϕ 2 σ P,c − σ z ,ΔT ) + 2[ − ]σ 0 ( 2 σ P,c − σ z ,ΔT ) − ( l A1 sinh2 ϕ A1 sinh2 ϕ A1 sinhϕ coshϕ sinhϕ − ϕ l )σ 02 2 A1 sinh ϕ
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+
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(
A 2 2 2(1 − ϕ cothϕ )(coshϕ − 1) + ϕ sinhϕ A 2 (ϕ cothϕ − 1)(coshϕ − 1) ϕ cothϕ − 1 2 ) +( ) + ]σ P,c A1 sinh2 ϕ A1 sinh2 ϕ sinh2 ϕ
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dU se = [(
(B.4)
(B.5)
The term Ufr is
U fr = ∫
Area
τ 0 ⋅ υ ( z ) dA
(B.6)
z
σ L − l + σ z , ∆T
L −l
Ef
υ ( z) = ∫ (ε final - ε initial )dz , ε final =
=
σ ( z) 1 2τ 0 ( L − z ) 1 2τ 0 l = ( + σ z ,∆T ) , ε initial = zzfd Ef Ef rf Ef rf
It is noted that the (B.7) is valid when the equation (B.8) is satisfied
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(B.7)
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(B.8)
The Ufr is obtained by substituting equation (B.7) into (B.6) , as follows 2πrf 1 1 1 τ 0 ( Ll 2 − L2l − ( L − l )3 + L3 ) + σ z , ∆T l 2 Ef 3 3 2
and therefore the work done against friction is solved, πrf2 2 Ef
σ 02l 2 + σ 0σ z , ∆T l dl
(B.10)
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dU fr =
(B.9)
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U fr =
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ACCEPTED MANUSCRIPT Figure Captions Figure 1. SEM micrographs of SiC fibers after push-out test for SiC(C)/Ti–6Al–4V (a and b) and SiC/Ti–6Al–4V (c and d) composites. (a) and (c) show the top face
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whereas (b) and (d) show the bottom face [35]. Figure 2. Idealized fiber push-out model.
Figure 3. Schematic representation of the axial stress in the fiber during push-out test.
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Figure 4. Critical applied stress necessary for crack growth, plotted as a function of
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interfacial toughness, for three crack lengths. Other variables are given in Table 1. Figure 5. Predicted dependence of the applied stress necessary for crack advance on the crack length. Other variables are given in Table 1.
Figure 6. Predicted dependence of the applied stress necessary for crack advance on the
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interfacial toughness. Plots are shown for the four thermal axial axial stresses. Other variables are given in the Table 1. Figure 7. Comparison of the critical applied stresses from Cases I and II as a function of
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the Γ for three frictional shear stresses. Other variables are given in Table 1.
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Figure 8. Comparison of the critical applied stresses from Cases I and II as a function of the Γ for three crack lengths. Other variables are given in Table 1.
Figure 9. The contributions of µ for the critical applied stress as a function of the Γ2 for the crack length 100 µm. Other variables are given in Table 1.
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ACCEPTED MANUSCRIPT Table 1. Parameters used to examine predictions of the analytical model Ef (GPa) 469 σP,c(MPa) -1300
Em(GPa) 110 σz,∆T(MPa) -500
υf 0.17 σr,∆T(MPa) -300
υm 0.26 τ0(MPa) -100
rf 71 v 0.35
rm 93 µ 0.3
L(µm) 500 Γ(Jm-2) 30
E(GPa)
υ
a (×10-6)
SCS-6 Sigma1240 Ti-6-4 Timetal 834 Timetal 21 Ti-24-11 Ti-15-3
469 400 115-30 115-70 98.2-40 110-43 88.3-44
0.17 0.21 0.36 0.3 0.35 0.26 0.29
4.0 4.0 10-7.3 11.24 9.5-6.8 11-9.0 11-9.7
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Reference [41] [33] [33] [41] [44,45] [46] [46]
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Material System
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Table 2．Thermo-elastic parameters of SiC and titanium alloy
l(µm) 20
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Table 3. The parameters of specimens and data obtained during the push-out test σP,c τ0 (Experimental) (Experimental) Material T rf L v [Reference] [Reference] System (℃) µm (µm) (µm) N MPa Sigma1240/Ti-6-4a 23 50 200 0.33 -1030 -127 Sigma1240/Ti-6-4a 600 50 200 0.33 -466 -28.7 SCS-6/Ti-6-4b 23 71 550 0.35 -1768 -448.6 SCS-6/Timetal 834c 23 71 400 0.4 -1958.5 -717.3 c SCS-6/Timetal 834 530 71 400 0.4 -587.5 -141 SCS-6/Timetal 21sd 23 70 530 0.35 -1559.9 -379.7 23 71 350 0.35 -1074.6 -198.9 300 71 350 0.35 -1123.9 -186.7 SCS-6/Ti-24-11e 400 71 350 0.35 -880.4 -128.3 650 71 350 0.35 -700 -93.3 815 71 350 0.35 -394.4 -21 400 71 350 0.35 -1212.7 -133 SCS-6/Ti-15-3e 700 71 350 0.35 -867.6 -105 815 71 350 0.35 -670.4 -35 The results obtained from: aRef. [33]; bRef. [47]; cRef. [41]; dRef. [21] and eRef. [46]
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Table 4. The thermal residual stresses and the interfacial toughness simulated by other workers and calculated by the equations (16i) and (16ii) of the present model. Γ2 Γ Γ1 Material σz,∆T σr,∆T (Equ.16i) (Equ.16ii) (Simulation) T(℃) -2 (MP) (MP) System (J m ) (J m-2) (J m-2) Sigma1240/Ti-6-4 23 -318a -255a 22.5 25 a Sigma1240/Ti-6-4 600 -21 -24.4a 5.7 6.3 SCS-6/Ti-6-4 23 -815.2 -346 55.3 58.1 52.5b SCS-6/Timetal 834 23 -839 -373 42.2 27.5 40b SCS-6/Timetal 834 530 -175 -73 6.4 6.2 5b SCS-6/Timetal 21s 23 -736 -348 42.6 44.8 50-70c 23 -563 -395 23.3 24.2 300 -312.5 -216 32 33.4 SCS-6/Ti-24-11 400 -134 -112 21 22 650 -58 -34 13.6 14.4 815 0 0 5.2 5.5 400 -237 -161 43.4 46.1 SCS-6/Ti-15-3 700 -63 -52 21.5 22.9 815 0 0 15.1 16.1 The results obtained from: aRef. [33]; bRef. [41]; cRef. [21]
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Figure1. SEM micrographs of SiC fibers after being pushed-out for SiC(C)/ Ti–6Al–4V (a and b) and SiC/Ti–6Al–4V (c and d) composites. (a) and (c)
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show the top face whereas (b) and (d) show the bottom face [35].
Figure 2. Idealized fiber push-out model.
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Figure 3. Schematic representation of the axial stress in the fiber during push-out
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testing.
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Figure 4. Critical applied stress necessary for crack growth, plotted as a function of interfacial toughness, for three crack lengths. Other variables are given in Table 1.
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Figure 5. Predicted dependence of the applied stress necessary for crack advance on
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the crack length. Other variables are given in Table 1.
Figure 6. Predicted dependence of the applied stress necessary for crack advance on
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the interfacial toughness. Plots are shown for the four thermal axial axial stresses. Other variables are given in the Table 1.
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Figure 7. Comparison of the critical applied stresses from Case I and Case II as a
function of the Γ for three frictional shear stresses. Other variables are given in Table
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1.
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Figure 8. Comparison of the critical applied stresses from Case I and Case II as a function of the Γ for three crack lengths. Other variables are given in Table 1.
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Figure 9. The contributions of µ for the critical applied stress as a function of the Γ2 for the crack length 100 µm. Other variables are given in Table 1.