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A micromechanics method for transverse creep behavior induced by interface diffusion in unidirectional fiber-reinforced metal matrix composites
Accepted Manuscript
A micromechanics method for transverse creep behavior induced by interface diffusion in unidirectional fiber-reinforced metal matrix composites Binbin Xu , Fenglin Guo PII: DOI: Reference:
S0020-7683(18)30388-3 https://doi.org/10.1016/j.ijsolstr.2018.09.024 SAS 10128
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
31 January 2018 2 August 2018 21 September 2018
Please cite this article as: Binbin Xu , Fenglin Guo , A micromechanics method for transverse creep behavior induced by interface diffusion in unidirectional fiber-reinforced metal matrix composites, International Journal of Solids and Structures (2018), doi: https://doi.org/10.1016/j.ijsolstr.2018.09.024
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Highlights
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Propose a new micromechanics method to analyze creep behavior induced by interface diffusion in metal matrix composites. Present an analytical solution for creep rate due to interfacial diffusion. Formulate a micromechanics model to estimate the stress field in the fiber accounting for interaction between fibers. Obtain overall creep strains and stress variation with time in fibers under constant external load by the incremental creep analysis procedures.
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A micromechanics method for transverse creep behavior induced by interface diffusion in unidirectional fiber-reinforced metal matrix composites
a
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Binbin Xua, Fenglin Guoa, b,*
School of Naval Architecture, Ocean and Civil Engineering (State Key Laboratory of Ocean Engineering), Shanghai Jiao Tong University, Shanghai 200240, China
b
Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240,
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China
Abstract
In this study, we present a new micromechanics method to predict the transverse creep rate
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induced by interface diffusion in unidirectional fiber-reinforced composites and evolution of overall creep strain under constant applied stress. An analytical solution for creep rate induced by
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interfacial diffusion, depending on the applied stress, fiber volume fraction and radius of the fiber, as well as the modulus ratio between the fiber and the matrix, is obtained. A micromechanics
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model is proposed to estimate the stress field in the fiber, which is related to the driving force for the interface diffusion, the normal stress on the interface. Comparison with finite element analysis
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shows that the present micromechanics model is of good accuracy, especially for high fiber volume fraction and large elastic modulus ratio between fiber and matrix. With the proposed
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micromechanics method based on the average field theory, the variation of overall creep strains and stresses with time in fibers under constant external load are analyzed by the incremental creep analysis procedures.
Keywords: Metal matrix composites (MMCs); Creep; Interface diffusion; Micromechanics
*
Corresponding author. E-mail: [email protected] (F. L. Guo). 2
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1. Introduction In recent years, particulate or fiber reinforced metal and intermetallic matrix composites (MMCs) have attracted considerable attention due to their good strength performance at high temperatures. The creep behavior of MMCs has been an essential topic of intense research. It is well-known that, in general, at modest temperatures the creep strength of metal and intermetallic matrix composites is better than that of the matrix alone because of the constraint of the
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reinforcement on the matrix. However, experimental results show that, at temperatures higher than about half of the melting temperature of the matrix, the composite strength is impaired and in some cases the strengthening imparted by the reinforcement is completely lost (Chun and Daniel, 1997; Nimmagadda and Sofronis, 1996). This creep behavior of composites is considered to be
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caused by diffusional relaxation and interface slip along the interface between the reinforcement and matrix. So far a great deal of effort has been devoted both theoretically and numerically in order to gain a better understanding on mechanisms of creep behavior of the composites. Chun and Daniel (1997) developed a micromechanics model based on the average field theory to predict
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transverse creep behavior in a unidirectional metal matrix composite. In their study, the matrix is considered as a creeping material following a Bailey-Norton power law, and creeping of the fibers
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and stress driven interfacial diffusion are neglected. Bullock et al. (1977) proposed an analytical model to predict creep rate of a Ni-Ni3AI-Cr3C2 eutectic composite by assuming that the fiber and
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matrix satisfy a power law of creep constitutive relationship, and have a homogeneous strain rate. McMeeking (1993) presented an asymptotic analysis for the power law creep of a matrix
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containing discontinuous rigid aligned fibers. Kim and McMeeking (1995) improved McMeeking’s model by incorporating the influences of interfacial slip and diffusion on the overall
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creep behavior of composites. Almost all analytical studies take the assumption of rigid reinforcements, which leads to inaccurate prediction of stress field along the interface and thus make it difficult to accurately assess the interface diffusion. In contrast with analytical investigations, numerical studies are capable of tackling general and complex problems, not only focusing on the overall creep behavior of composite based on the creep law of each constituent, but also paying attention to simulating the stress and strain relaxation processes, which bring in in-depth understanding to the mechanism of interfacial slip and diffusion induced creep behavior of the composites. Park and Holmes (1992) studied the 3
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tensile creep and creep-recovery behavior of fiber reinforced ceramic composites using finite element method with rectangular unit cells. The influences of interface condition, fiber volume fraction, as well as processing-related residual stresses were all considered (Park and Holmes, 1992). Later, Sofronis and McMeeking (1994) investigated the creep behavior of particulate composite materials by coupling the diffusion and slip along the interface with creep deformation of the matrix. It is found that normal stress gradients on the interface will induce diffusional flow
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along the interface accompanied by slip of the matrix over the reinforcement, which leads to the relaxation of the constraint. Finite element analysis on unit cell models showed that either diffusional relaxation or slip may knock down the creep resistance of the composite to levels even below the matrix strength (Sofronis and McMeeking, 1994). However, numerical studies are
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limited to fixed computational parameters and it is difficult to draw a broadly applicable conclusion based on the results of a small number of case studies. Thus, it is important to develop analytical solutions which are able to reveal the inherent characteristics of creep behavior caused by diffusional relaxation and interface slip.
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Experimental studies indicate that composites with micron- or nano-sized inclusions exhibit size effect on the overall effective elastic and plastic properties, which cannot be adequately
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captured by classical continuum mechanics theory. Discrete dislocation plasticity analysis has thus been conducted to investigate deformation of metal matrix composites and superalloys with
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micron- or nano-sized constituents at high temperatures (Shishvan et al., 2017a, 2017b; Shishvan et al., 2018). These studies show that elastic fields of dislocations in the matrix will affect the
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stress-driven interfacial diffusion in composites. In addition, the decrease in the strength of the composites due to interfacial diffusion is a result of a change in the dislocation structures within
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the matrix that, in turn, leads to a reduction in the matrix strength (Shishvan et al., 2017b). Metal matrix composites, in particular continuously reinforced MMCs, have demonstrated
potential for high temperature propulsion and airframe applications in aerospace industry because of their brilliant properties at elevated temperature in the fiber direction. Unfortunately, the transverse creep resistance and the creep-rupture life are much lower than those in the fiber direction due to the weak nature of the fiber/matrix interfacial bond and interfacial diffusion (Bednarcyk and Arnold, 2002; John et al., 1996). Therefore, a thorough investigation on the transverse creep performance is necessary in order to take full advantage of this type of metal 4
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matrix composites in engineering applications. Wang and Weng (1992) developed a local-field theory based on the combination of Mori-Tanaka method and the analytical solution of a three-phase cylindrically concentric solid to study the evolution of stress distribution in the matrix and overall creep strain of a fiber-reinforced metal-matrix composite. Their local-field theory had shown to provide a more accurate estimate for the overall transverse creep of the composite than the simpler Mori-Tanaka method. Chun and
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Daniel (1997) proposed a micromechanical model based on Mori-Tanaka average field theory to predict transverse creep deformation in a unidirectional metal matrix composite. Their results showed that the composite creep strain rate was less sensitive to the stress amplitude and temperature than that of the unreinforced matrix material. In order to elucidate the damage
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mechanisms under transverse creep loading of unidirectional metal matrix composites, Carrère et al. (2002) simulated the creep deformation process on a representative unit cell using the finite element method with coupling effect between matrix creep and interface damage taking into account. Numerical results showed that the global response of the composite subjected to a
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transverse creep loading strongly depends on the local interfacial behavior. Nevertheless, none of above mentioned researches on transverse creep of MMCs took the interfacial diffusion induced
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creep into consideration, which could be significant in high temperature applications. Recently, several researchers obtained the stress field in the fiber by Eshelby solution and
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concluded that the interface slip only has a short range effect compared to diffusional mass transport in unidirectional fiber reinforced composites (Li et al., 2014; Li and Li, 2012). They
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presented an analytical solution for the transverse creep rate contributed by interface diffusion. Their researches clearly visualized the roles of some important parameters for the interface creep,
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such as the fiber size and volume fraction. However, their method of calculating the normal stress gradient along the interface is only applicable for small fiber volume fraction. In unidirectional fiber reinforced composites, the creep behavior exhibits strong anisotropy
and creep rate is very low or negligible in the longitudinal direction but more pronounced in the transverse direction. Because the state of stress in the matrix is highly heterogeneous under transverse loading, it is quite difficult to obtain exact predictions of average creep behavior (Wang and Weng, 1992). A simple micromechanics model would be very desirable. In the present study, the normal stress gradient along the interface, which induces diffusional mass transport, is 5
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estimated by averaging results of self-consistent scheme (Hill, 1965) and the Mori-Tanaka method (Mori and Tanaka, 1973). Comparison of the proposed method with finite element analysis shows that the present method is able to give prediction of average stress in fiber with reasonable accuracy, especially for high fiber volume fraction and large contrast between fiber and matrix moduli. We also present an analytical solution of creep strain rate, which is formally similar to that of Coble creep in polycrystalline material (Coble, 1963) and creep rate due to diffusional mass
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transfer given by Mori et al. (1997). Then the deformation of fibers and creep behavior of metal matrix composites are well elucidated within the framework of the micromechanics method. With the proposed micromechanics method based on the average field theory, the accumulated creep strain and stress transfer occurring in fibers under constant external load are analyzed by the
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incremental creep analysis procedures. This study may bring new insights to understanding of transverse creep behavior and designing composite materials of higher creep resistance.
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2. Diffusional creep model
In numerical studies, the unit cell model is generally preferred if fiber arrangement in
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composites is assumed as a hexagonal or rectangular array (Berton et al., 2006; Mori et al., 1997; Nimmagadda and Sofronis, 1996). However, since this study is concerned with analytical analysis
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of transverse creep behavior of MMCs by micromechanics theory, it is not necessary to adopt the unit cell model in the present work. The fiber reinforcements are simulated as long parallel
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cylinders in the longitudinal direction. As depicted in Fig.1, a fiber with a radius of is embedded in matrix subjected to a general in-plane external stress σ 0 . The gray circle in Fig. 1
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stands for the cross-section of the long fiber. For simplicity, matrix and fibers are assumed to be homogeneous and isotropic and the interface between them is perfectly bonded. Upon loading the normal stress gradient along the interface causes matter flow from the sides to the poles of the fiber, resulting in creep deformation in the loading direction.
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σ0
y Vn
j
x
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matrix
σ0
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Fig. 1. A schematic diagram of interface diffusion in unidirectional fiber-reinforced composites.
The chemical potential of the atoms on the interface is (Li and Li, 2012)
0 s n ,
0 is the reference value of the potential, which is considered to be constant for a given
material.
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where
(1)
s and are the interface energy and the atomic volume respectively. is the
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curvature of the interface, a constant for a circular interface.
n is the normal stress acting on the
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interface, positive for tension and negative for compression. Besides, strain energy term has been taken to be negligible (Li and Li, 2012; Li et al., 2010; Sofronis and McMeeking, 1994).
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The volumetric flux per unit length crossing the interface is given by (Herring, 1950)
j
Di i Di i n , kT s kT s
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where Di is the interface diffusion coefficient,
(2)
i is an effective interface thickness, k is
the Boltzmann constant, T is the absolute temperature, and s is the arc length of the interface. The law of mass conservation along the interface requires
j Vn 0 , s
(3)
where Vn is the rate of accumulation or depletion of matter on the interface, positive for accumulation and negative for depletion. The above equation is the governing equation of the diffusional mass flow, which provides a constitutive type of relationship between stress and 7
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velocity on the interface boundary. Moving Vn to the right side of the equation, we have
Di i 2 n j Vn . s kT s 2
(4)
Thus, the creep rate of the fiber along the radial direction is given by
2Vn D 2 n i i . 2 kT s 2
(5)
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The crux of obtaining the diffusional creep rate is formulation of an appropriate micromechanics model to accurately estimate the stress distribution at the interface.
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3. Estimation of average stresses in fibers
Micromechanical analyses have been widely used to predict the effective properties (elasticity, plasticity, conductivity, etc.) of composites of the matrix-inclusion type. So far, a number of theoretical micromechanics methods have been proposed based on the pioneering work
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of Eshelby (Eshelby, 1957), including Mori-Tanaka method (Mori and Tanaka, 1973), self-consistent scheme (SCS) (Hill, 1965), differential scheme (DS) and double-inclusion model
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(Hori and Nemat-Nasser, 1993) as well as the interaction direct derivative estimate (IDD) (Zheng and Du, 2001), among others. Besides, there are two micromechanics models which possess
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mathematical rigor in their elasticity formulation and are applicable to high reinforcement volume fraction. They are composite spheres (cylinders) model and generalized self-consistent method
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(GSCM) (Christensen, 1998; Christensen and Lo, 1979). In this work, we are interested in how the average stress in inclusions could be estimated
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accurately by micromechanics methods. However, unlike the effective properties of composites, the stress in inclusions is much more difficult to estimate with decent accuracy, particularly for high fiber volume fraction and large elastic modulus ratio between fiber and matrix. It is generally recognized that Mori-Tanaka method tends to underestimate overall effective elastic moduli whereas self-consistent scheme tends to overestimate overall effective elastic moduli when the inclusion is stiffer than the matrix (Christensen, 1990). Results of finite element analysis in this study indicate that Mori-Tanaka method tends to underestimate the average stress in the loading 8
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direction in fibers while self-consistent scheme tends to overestimate the average stress in fibers for metal matrix composites. Accordingly, taking average of results of the Mori-Tanaka method and the self-consistent scheme may give a reasonable estimate for average stress in the fibers. The accuracy of this approach will be discussed later in this section for different fiber volume fractions and ratios of Young’s modulus between fiber and matrix. In Mori-Tanaka method, the ellipsoidal inclusion is placed within a homogenous matrix
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which had been subjected to the as-yet-unsolved average stress of the matrix before the inclusion was embedded. The average stress-strain relationship for the matrix is expressed as
σm σ0 σm Cm (ε0 εm ) ,
(6)
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where Cm is the stiffness tensor of the matrix, σ 0 is applied constant external stress, ε 0 is the elastic strain corresponding to the applied stress σ 0 in an equivalent body without reinforcement,
σm and ε m are respectively matrix stress and strain disturbances due to the presence of reinforcement. Similarly, with the help of Eshelby’s equivalent inclusion principle, the stress in the
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fiber can be expressed as
σ f σ0 σ f
(7)
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C f Cm1 (σ 0 σ m ) ε f ε c , Cm Cm1 (σ 0 σ m ) ε f ε c ε*
where C f is the stiffness tensor of the fiber, σ f and ε f are respectively the disturbance
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stress and strain due to the presence of the inclusion and creep strain, εc is the creep strain in the
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inclusion and ε* is a fictitious eigenstrain which is introduced to relate the present problem to the equivalent inclusion problem. The relationship between the disturbance strains ε f and ε* is expressed as
ε f S(ε c ε* ) , where
(8)
S is the Eshelby’s tensor. The equilibrium condition with Eqs. (6) and (7) gives (1 f )σ m fσ f 0 ,
where f is the fiber volume fraction. 9
(9)
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Substituting Eq. (8) into Eq. (7) and making use of the equilibrium condition Eq. (9), the fictitious eigenstrain ε* can be solved as 1
ε* f (S I) S (C f Cm )1 Cm Cm1σ0 (1 f )(S I)εc ,
(10)
where I is the fourth-order symmetric identity tensor. From Eq. (7) and Eq. (9), the stress
σ f (1 f )Cm (S I)(ε* ε c ) σ m fCm (S I)(ε* ε c )
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disturbances in the fiber and matrix can be represented by ε* as
.
(11)
Substituting Eq. (11) into Eq. (7), the stress in the fiber can be expressed by
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σ f C f Cm1 (σ 0 σ m ) ε f ε c C f Cm1σ 0 f (S I)(ε* ε c ) S(ε* ε c ) ε c .
(12)
C f Cm1σ 0 (1 f )(S I)ε c [(1 f )S fI] *
Substituting Eq. (10) into this equation and after a long process of algebraic manipulation, the stress in the fiber can finally be given by
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σ Mf T C f ( A I ) 1 Cm1σ 0 (1 f )(S I)ε c ,
(13)
where A (1 f )S fI Cm (C f Cm ) . If there is no creep strain ( c 0 ) and volume
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fraction f 0 , Eq. (13) can be further reduced to the Eshelby solution 1
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σ Eshelby I Cm (I S)(Cf 1 Cm1 ) σ 0 . f
(14)
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On the other hand, in the self-consistent scheme, the applied loads and the interaction with other inhomogeneities are accounted for by assuming the inclusion is placed within a
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homogeneous matrix with as-yet-unknown effective properties, which had been subjected to uniform strain before the inclusion was embedded. It should be noted that the self-consistent scheme actually yields implicit equations for the effective properties. In order to obtain an explicit solution of the stress in the inclusion, we use Mori-Tanaka method to estimate the effective properties of the composite. The derivation process of stress expression is similar to the Eshelby solution, and the difference is that the inclusion is embedded in an effective medium of stiffness
C . Thus, the equivalent inclusion principle equation gives
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C f C1σ 0 ε ε c C C1σ 0 ε ε c ε*
(15)
where ε is the disturbance stress due to the presence of inclusions and creep strain, it can be expressed as
ε S(εc ε* )
(16)
Introducing Eq. (16) into Eq. (15), the eigenstrain ε* can be solved 1
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ε* C(S I) C f S (C f C)C1σ 0 C f εc εc
(17)
Once the eigenstrain ε* is known, the stress in the inclusion (fiber) can be computed, which is finally expressed in the following form after a series of mathematical operations
1
1
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σ SCS C f SC1 (C f C) I C1σ0 I (S I)1 SC1C f εc , f
(18)
where the effective stiffness tensor of the composite C is estimated through Mori-Tanaka method as
C (1 f )Cm fC f T (1 f )I fT
1
1
(19)
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with T I SCm 1 (C f Cm ) . From the preceding equation, the effective shear and
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in-plane bulk modulus of the fiber reinforced composite are
m 1
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f (1 f )(3 4 m ) / 4(1 m ) m / ( f m )
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f K K m 1 (1 f ) / 2(1 m ) K m / ( K f K m )
,
(20)
subscripts f
m denote the fiber and the matrix, respectively. The effective Young’s
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where , K and and
are the shear modulus, bulk modulus and Poisson’s ratio, respectively. The
modulus E and Poisson’s ratio v are related to shear and bulk moduli by
E (3
) K . 1 (1 ) 2 K
(21)
It is noted that the effective elastic modulus E and Poisson’s ratio obtained by Eq. (20) are the in-plane properties of the composite. Actually, a unidirectional metal matrix composite 11
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possesses transverse isotropy and the plane of isotropy is perpendicular to the direction of fiber. This property enables one to describe the in-plane properties of the composite by Young’s modulus and Poisson’s ratio. Taking the average of Eq. (13) and Eq. (18) gives
average f
1 M T ( f SCS ). f 2
(22)
stress components in the above equation can be given by
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This is the final expression of the estimated fiber stress. If the creep strain in the fiber is zero, the
x x 0 , y y 0 , xy 0 , where
(23)
0 is the applied far-field stress and expressions of coefficients x and y are
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presented in Appendix A for brevity.
In order to check the accuracy of Eq.(22), detailed FE-analyses are performed, in which the fiber packing of composites is approximated by a simple rectangular array (see Fig.2) since optical micrograph shows that the fiber packing in unidirectional fiber-reinforced composites can
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typically be idealized as either hexagonal or rectangular array (Park and Holmes, 1992). Considering the micromechanical analysis is meant to obtain effective properties of composites
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containing a large number of inclusions, the number of fibers in the FEM model should be large enough. When the overall size and fiber volume fraction of a finite element model are kept
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constant, the greater the number of fibers, the smaller the fiber size and vice versa. Accordingly, three different models are established, namely model (1), model (2) and model (3), in which the
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ratio of the cross-sectional area of a single fiber to the entire model is 1%, 5‰ and 1‰, respectively. These models are stressed in tension in the y-direction and appropriately meshed.
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According to Eshelby solution (Eshelby, 1957), the stress is uniform in a single elliptical inclusion embedded in an infinite matrix and the result has been proved to be mathematically rigorous. Although in reality the stress field in fibers is obviously heterogeneous, finite element results show that, if the influence of boundaries is ignored, the stress fluctuation in fibers is minimal in a wide range of fiber volume fraction, as shown in Fig. 2.
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Fig. 2. Finite element model (model (3)) of rectangularly arrayed fiber-reinforced composites with 28.9% fiber volume fraction consists of a total of 289 ( 17 17 ) fibers. The fluctuation of stress component in fibers in the loading direction y is around 7.4% ( m f 0.2 , 10 ).
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The accuracy of Eq. (22) is verified by FEA under various values of fiber volume fraction and modulus ratio of fibers to matrix. Figs. 3 and 4 present comparisons of average stress values in
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fibers calculated by FEA, the present micromechanics method, Eshelby solution, self-consistent scheme and Mori-Tanaka method. FEA results show that the relative size or the number of fibers
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has a slight effect on the average stress of fibers, which increases slightly with rising number of fibers when the fiber volume fraction is constant. This effect gradually decreases as the number of
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fibers increases and it is considered herein that the size of fibers in the model (3) is sufficiently small since the average stress variation of fibers is very small and neglected when the number of
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fibers is further increased. One can see that, when the modulus ratio between fibers and matrix is
E f Em 10 , the stresses in fibers calculated from Eq. (22) are very close to those of
FE-analysis for varying fiber volume fraction from 0% up to 50%, and the relative error with the results of FEM model (3) is less than 0.8%. As modulus ratio and fiber volume fraction increase, the accuracy of self-consistent scheme and Mori-Tanaka method will drop as well. Fig. 4 shows that even when the fiber volume fraction is 48.4% and modulus ration is up to 12, the estimate given by the present method is still in good agreement with that of FEA with very small error. 13
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Numerical results obviously show that the micromechanics model proposed in the present study is more accurate than direct use of Eshelby solution. Therefore, the average stress obtained by Eq. (22) gives a reasonable estimate of stresses in fibers.
1.35
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1.40
10
1.30
FEM-model (1) FEM-model (2) FEM-model (3) self-consistent scheme M-T method present method Eshelby solution
1.25
1.20
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Normalized stress y
1.45
1.15 0
10
20
30
40
50
Volume fraction (%)
Fig. 3. Comparison of normalized stress y 0 in the fiber calculated from Eq.(22), FE-analyses, Eshelby
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solution, self-consistent scheme and Mori-Tanaka method for varying fiber volume fraction ( m f 0.2 ,
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10 ).
1.6
Normalized stress y/
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1.4 1.2 1.0 0.8
FEM-model (3) self-consistent scheme Eshelby solution M-T method present method
0.6
f=0.484 0.4 0.2 0.0 0
2
4
6
8
10
12
Elastic modulus ratio
Fig. 4. Comparison of normalized stress y 0 in the fiber calculated from Eq.(22), FE-analyses, Eshelby solution, self-consistent scheme and Mori-Tanaka method for varying
f 48.4% ). 14
in a range of 0–12 ( m f 0.2 ,
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To further examine the accuracy of Eq. (22) we write average stress in the fiber in the following weighted average form
f (1 w) Mf T w SCS , f
(24)
where the weight w is a function of the fiber volume fraction f and the elastic modulus ratio
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between fibers and matrix. It can be determined by making the value of f in Eq. (24) fit the FEA results. An approximate formula of polynomial fitting for the weight w is given below
w( f , ) (1.0200305 4.2943 f 16.6368 f 2 15.8651 f 3 )
(1.08453 0.15671 0.0132665 2 0.00040065 3 )
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where f [0.064, 0.484] and [5,12] .
(25)
For a typical value 10 , the weight factor w changes from 0.445 to 0.546 when the fiber volume fraction varies from 6.4% to 48% . It can be seen from Fig. 3 that both Mori-Tanaka method and self-consistent scheme will approach the Eshelby solution for the case of
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small fiber volume fraction.
By micromechanics theory, the average stress in inclusions or fibers is related to the overall
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effective stiffness tensor of the composite by
1 1 (Ci Cm1 )1 (C1 Cm1 )σ 0 . f
(26)
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σi
Derivation of Eq. (26) is given in Appendix B. As mentioned earlier in this section, Mori-Tanaka
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method tends to underestimate overall effective elastic moduli whereas self-consistent scheme tends to overestimate overall effective elastic moduli when the inclusion is stiffer than the matrix.
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Therefore, from Eq. (26) we deduce that the actual average stress in the fiber lies between 1 f
(Ci1 Cm1 )1 (CM1T Cm1 )σ0 and
1 f
1 (Ci1 Cm1 )1 (CSCS Cm1 )σ0 for metal matrix composites.
Based on these arguments, we conclude that taking average of results of the Mori-Tanaka method and the self-consistent scheme provides a simple and reasonable approach to estimate average stress in inclusions or fibers in view of the complexity of fiber distribution and interaction in MMCs. 4. Transverse creep rate 15
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As mentioned before, in the fiber-reinforced composite with a rectangular array of fiber packing, the stresses in fibers are approximately uniform and can be estimated analytically even though the stress distribution in matrix is complicated. Thus, stress components in fibers in a polar coordinate system can be expressed by
x y x y cos 2 2 2 y x y x cos 2 , 2 2 y r x sin 2 2 r
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where
(27)
x and y are stress components in the directions x and
y as presented in Eq.(23),
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which are also the principal stresses in fibers for this problem. r r is just the normal stress n along the interface between the fiber and the matrix.
From Eqs. (27) and (5) and noting that ds d , we have
2 Di i ( x y ) cos(2 ) 3kT . 2 Di i 3 x y 0 cos(2 ) kT
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r
(28)
Then, it follows from Eq. (28) that
2 Di i x y 0 3kT . 2 Di i 3 x y 0 kT
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x r 0
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y r 90
(29)
It is obvious that the creep strain rate in the 45 direction is zero by Eq. (28), which implies
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xy =0. The average creep strain rate of the fiber can be expressed in the following form
where
x ε c 0 0
0
y 0
0 0 , 0
(30)
x and y are components of creep strain rate in directions of x-axis and y-axis,
respectively. Eqs. (28) and (29) indicate that the creep rate of the fiber depends on the fiber volume 16
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fraction, the radius of the fiber, and material properties of the fiber and matrix as well as the applied stress. It should be noted that we have taken account of interaction between fibers in the process of deriving creep rate expression. As a result, the expression of creep rate obtained in the present study is expected to be valid for a wide range of fiber volume fraction up to 50%. Although the creep rate component
y is formally similar to those of previous works, we also
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note that the creep rate in Eq. (30) is presented in the form of tensor and is essentially different from the previous works (Coble, 1963; Li and Li, 2012; Wakashima et al., 1990), which only consider the creep rate in the loading direction. However, all these results predict linear dependence of creep rate on the applied stress.
So far, we have derived the tensor form of creep rate. As can be seen from Eq. (28), the creep
r is anti-symmetrical with respect to the direction 45
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rate component
and
r (45 ) 0 , which means there is no creep occurring in the fiber along the diameter in the 45 direction and the diffusional creep deformation in fibers is incompressible. The illustration
y
CE
PT
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of fiber deformation caused by creep due to interfacial diffusion is schematically shown in Fig. 5.
45°
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x -45°
Fig. 5. Illustration of stress-directed diffusional mass transfer along the interface between fiber and matrix in the 17
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composite
5. Transverse creep behavior under constant stress Metal-matrix composites (MMCs) have found many applications in aircraft and automotive industries as well as civil engineering, as structural components due to their excellent physical and mechanical properties at high temperatures. Creep behavior is usually a major concern when
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evaluating safety of structural components made of metal-matrix composites operating at high temperatures or high stress level. For example, creep of a turbine blade will cause the blade to contact the casing, resulting in failure of the blade. Therefore, the creep of MMCs should be precisely analyzed to guarantee structural safety. When an external constant stress is applied, the
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fiber and matrix are subjected to an initial stress distribution upon equilibrium. The corresponding initial stress gradient along the interface causes diffusional mass transport, which is generally assumed as plastic strain, in turn, disturbs the initial stress distribution. In this study, we investigate the creep behavior due to interfacial diffusion in the composite, and the fiber and
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matrix are assumed to deform elastically due to the applied stress and the interfacial diffusion. Eventually, the transverse creep behavior can be predicted by incremental analysis procedures.
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The total average strain of the composite can be obtained from Eqs. (22) and (30) t
ε(t ) (1 f )Cm1σ m (t ) fCf 1σ f (t ) f εc dt ,
(31)
t0
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where t is time and σ m (t ) satisfies (1 f )σ m (t ) fσ f (t ) σ0 . In the process of creep
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deformation, the creep strain due to the diffusional mass transfer at the inclusion/matrix interface, which can be introduced into the fiber as eigenstrain, may affect the stress distribution in the fiber
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and matrix. Although it is hard to obtain explicit solution for this problem, we can calculate the overall average strain and the average stress in the fiber at any time t by incremental analysis procedures as long as the initial creep rate is known. At initial moment t t0 , no creep occurs in the composite. The stress fields in the fiber and
matrix are respectively
1 1 σ f (t0 ) C f ( A I) 1 Cm1 SC1 (C f C) I C1 σ 0 . 2 σ m (t0 ) [σ 0 fσ f (t0 )] / (1 f )
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(32)
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For a small time increment dt , we assume that the creep rate is constant and the creep strain induced by interface diffusion can be expressed as
c (t )dt , then the stress fields in matrix and
fiber become
1 C (1 f )(S I )(A I ) 2
σ f (t dt )
1 1 C f ( A I) 1 Cm1 SC1 (C f C) I C1 σ 0 2 1
f
I (S I )1 SC1C f
t ε c dt ε c (t )dt t 0
(33)
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and
1
σ m (t dt ) [σ0 fσ f (t dt )] / (1 f ) . The updated total strain in the composite is
(34)
(35)
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t ε(t dt ) (1 f )Cm1σ m (t dt ) fCf 1σ f (t dt ) f εc dt εc (t )dt . t 0
These incremental calculations can be carried out with the help of mathematical software like Mathematica or Matlab.
In order to examine the creep behavior of composites induced by interfacial diffusion, the
(t ) / (t0 ) and (t ) / (t0 ) , is plotted as a function of normalized time
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direction,
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normalized transverse creep strain and the creep strain rate of the composite in the loading
t 2Diit / 3kT for different fiber volume fraction f and elastic modulus ratio in
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Figs. 6 and7. The applied external stress is 20Mpa and the elastic modulus of matrix is 50Gpa. The
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Poisson’s ratios of fiber and matrix are both 0.2.
2.8 2.6 2.4
1.0
=10
0.1 =0.2 =0.3 =0.4
0.8
2.2 0.6
=0.1 =0.2 =0.3 =0.4
ε(t ) 2.0 ε(t 0 ) 1.8
ε( t ) ε( t0 ) 0.4
1.6 1.4
0.2
1.2 1.0 0
5
10
15
20
t (0.0025)
19
25
30
35
0.0 40
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Fig. 6. The normalized creep strain (solid lines) and the corresponding strain rate (dashed lines) as a function of the normalized time t 2Di i t / 3 kT for composites with different fiber volume fractions at fixed 10 and f m 0.2 .
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2.0
1.0
=0.3
=2 =5 =10 =12
2.2
0.8
1.8
0.6
=2 =5 =10 =12
ε( t ) 1.6 ε( t 0 ) 1.4
ε( t ) ε( t0 )
0.4
0.2
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1.2
1.0 0
5
10
15
20
25
30
35
0.0 40
t (0.0025)
Fig. 7. The normalized creep strain (solid lines) and the corresponding strain rate (dashed lines) as a function of
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the normalized time t 2Di i t / 3 kT for composites with different elastic modulus ratios at fixed f 0.3
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and f m 0.2 .
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As can be seen from Figs. 6 and 7, both fiber volume fraction and elastic modulus of fiber have a large impact on creep resistance of the composite material. As far as the creep due to
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interfacial diffusion is concerned, higher volume fraction or elastic modulus of the fiber results in larger total creep strain of the composite, although this favors the overall elastic properties. It is
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observed that fiber volume fraction has a bigger effect on creep strain than the elastic modulus of fiber. This is because larger fiber volume fraction means that the composite contains larger area of interface between the fiber and the matrix, leading to more interfacial diffusion. The maximum creep strain in the loading direction, which occurs in the case 10 and f 0.4 , is over 2.6 times the initial elastic strain. Besides, these results show that the creep due to interfacial diffusion will mainly contribute to the primary creep regime as the corresponding creep strain rate eventually approaches zero. The change in the stress field of the composite is another major concern. Since stress field in the matrix is highly heterogeneous, only the stresses in the fiber, 20
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where the gradual stress transfer occurs, are analyzed. The two perpendicular stress components in the fiber are shown in Fig. 8 and Fig. 9 in various fiber volume fractions and elastic modulus ratios. Figs. 8 and 9 show that the stress component the elapse of time while the stress component
y in the loading direction decreases with
x changes just in the opposite way. This
phenomenon indicates that the normal stress gradient and shear stress along the interface will
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decrease gradually and explains the reason why the creep rate induced by interfacial diffusion eventually drops to zero. It is also found that the stress in fiber is more sensitive to the elastic modulus ratio than the fiber volume fraction. These results may bring new insights to
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understanding of diffusional creep behavior.
1.4
Stress component y
1.2 1.0
σ0
0.8
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σ f (t )
0.6 0.4
Stress component x
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0.2 0.0
5
10
15
20
25
30
35
40
t (0.0025)
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0
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Fig. 8.The normalized stress in fibers as a function of the normalized time t 2Di i t / 3 kT for the
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composites with different fiber volume fractions at fixed 10 and f m 0.2 .
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1.4
Stress component y
1.2
1.0 0.8
σ f (t ) 0.6
Stress component x
0.4 0.2 0.0 0
5
10
15
20
t (0.0025)
25
30
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σ0
35
40
Fig. 9. The normalized stress in fibers as a function of the normalized time t 2Di i t /
6. Discussions and concluding remarks
f m 0.2 .
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composites with different Young’s module ratios at fixed f 0.3 and
3 kT for the
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In this study, we have developed a new micromechanics method to predict the transverse creep rate induced by interface diffusion in unidirectional fiber-reinforced composites and
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evolution of overall creep strain under constant applied stress. The driving force for the interface diffusion is the normal stress acting on the interface, which can be estimated by micromechanics
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methods. The proposed micromechanics model, which takes the average of the results of self-consistent scheme and Mori-Tanaka method, enables one to give estimate of the stress field in
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the fiber of fiber reinforced metal matrix composites with reasonable accuracy. Comparison with finite element analysis shows that the present method is better than direct use of Eshelby solution,
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especially for high fiber volume fraction and large contrast between fiber and matrix elastic moduli. For the case of small fiber volume fraction (below 10%), results of Eshelby solution, Mori-Tanaka method and self-consistent scheme are close to each other. When the fiber volume fraction is larger than 20%, the difference among these methods is getting obvious. For composites with small fiber volume fraction, we still can use Eshelby solution to quantify major features of interfacial diffusion creep in metal matrix composites (Li and Li, 2012). 2012) For composites with large fiber volume fraction, the micromechanics method proposed in this study 22
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offers a better choice for investigation of interfacial diffusion creep since it is able to give more accurate estimate of stress in fibers. An analytical solution of creep rate induced by interfacial diffusion, depending on the applied stress, volume fraction and radius of the fiber, as well as the modulus ratio between the fiber and the matrix, is presented. The deformation of fiber and composite as well as evolution of overall creep strain can be well described by this micromechanics method. With the proposed
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micromechanics method based on the average field theory, the creep strains and stress transfer under constant external force occurring in fibers are analyzed by the incremental creep analysis procedures. The methodology and results of this study may bring new insights to understanding of
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the creep behavior and designing composite materials of higher creep resistance.
The interfacial diffusion and interfacial slip are generally regarded to be related to each other. Petersonet al. (2003) proposed that interfaces may slide via interface diffusion controlled diffusional creep, which had been validated experimentally. On the other hand, interface slip may
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make stress field along the interface redistribute, and in turn, affect interface diffusion driven by the normal stress along the interface. Li Y and Li Z (2012) mentioned that the interface slip is
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limited to the local regions of the interface and only has a short range effect. Besides, the effect of interface slip can be taken into account by incorporating it into the creep of matrix. Therefore, the
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present micromechanics method without taking interface slip into consideration is applicable when we only focus on the creep due to interfacial diffusion.
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In this work, we only studied the creep due to mass diffusion at interface between the fiber and matrix and did not consider the creep of matrix. Evidently, the creep induced by the interfacial
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diffusion is always coupled with matrix creep in the real metal matrix composite materials. In general, the matrix creep could make stress in the composite redistribute, and will, in turn, affect the interfacial diffusion. On the other hand, the development of creep strain in fibers due to interface diffusion will counteract the applied stress and thus make the average stress in the matrix increase. Only after the matrix creep and the interface creep are thoroughly investigated individually, would it be possible to fully understand the intricate interaction between them. This is beyond the scope of the present study and will be the task for future research.
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Acknowledgement
The authors gratefully acknowledge the financial support by the National Natural Science Foundation of China through Grant No. 11272206.
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References
Bednarcyk, B.A., Arnold, S.M., 2002. Transverse tensile and creep modeling of continuously reinforced titanium composites with local debonding. International Journal of Solids & Structures 39, 1987-2017.
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Berton, J.R., Durney, D.W., Wheeler, J., Ford, J., 2006. Diffusion-creep modelling of fibrous pressure-shadows. Tectonophysics 425, 191-205.
Bullock, E., McLean, M., Miles, D., 1977. Creep behaviour of a Ni-Ni3Al-Cr3C2 eutectic composite. Acta Metallurgica 25, 333-344.
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Carrère, N., Kruch, S., Vassel, A., 2002. Damage Mechanisms in Unidirectional SiC/Ti Composites under Transverse Creep Loading: Experiments and Modeling. International Journal
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of Damage Mechanics 11, 41-63.
Christensen, R.M., 1990. A critical evaluation for a class of micro-mechanics models. Journal of
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the Mechanics & Physics of Solids 38, 379-404. Christensen, R.M., 1998. Two Theoretical Elasticity Micromechanics Models. Journal of Elasticity
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Christensen, R.M., Lo, K.H., 1979. Solutions for effective shear properties in three phase sphere
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and cylinder models. Journal of the Mechanics & Physics of Solids 27, 315-330. Chun, H., Daniel, I., 1997. Transverse creep behavior of a unidirectional metal matrix composite. Mechanics of materials 25, 37-46.
Coble, R.L., 1963. A Model for Boundary Diffusion Controlled Creep in Polycrystalline Materials. Journal of Applied Physics 34, 1679-1682. Eshelby, J.D., 1957. The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 241, 376-396. 24
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Herring, C., 1950. Diffusional viscosity of a polycrystalline solid. Journal of applied physics 21, 437-445. Hill, R., 1965. A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids 13, 213-222. Hori, M., Nemat-Nasser, S., 1993. Double-inclusion model and overall moduli of multi-phase composites. Mechanics of Materials 14, 189-206.
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John, R., Khobaib, M., Smith, P.R., 1996. Prediction of creep-rupture life of unidirectional titanium matrix composites subjected to transverse loading. Metallurgical & Materials Transactions A 27, 3074-3080.
Kim, K.T., McMeeking, R.M., 1995. Power law creep with interface slip and diffusion in a
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composite material. Mechanics of materials 20, 153-164.
Li, Y., Li, S., Li, Z., 2014. Interface diffusion-induced creep and stress relaxation in unidirectional metal matrix composites under biaxial loading. Mechanics of Materials 76, 20-26. Li, Y., Li, Z., 2012. Transverse creep and stress relaxation induced by interface diffusion in
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unidirectional metal matrix composites. Composites Science and Technology 72, 1608-1612. Li, Y., Li, Z., Wang, X., Sun, J., 2010. Analytical solution for motion of an elliptical inclusion in
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gradient stress field. Journal of the Mechanics and Physics of Solids 58, 1001-1010. McMeeking, R.M., 1993. Power law creep of a composite material containing discontinuous rigid
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aligned fibers. International journal of solids and structures 30, 1807-1823. Mori, T., Huang, J., Taya, M., 1997. Overview no. 126 Stress relaxation by plastic flow, interfacial
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sliding and diffusion in an inclusion bearing material. Acta materialia 45, 429-438. Mori, T., Tanaka, K., 1973. Average stress in matrix and average elastic energy of materials with
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Park, Y., Holmes, J., 1992. Finite element modelling of creep deformation in fibre-reinforced ceramic composites. Journal of materials science 27, 6341-6351. Peterson, K.A., Dutta, I., Chen, M.W., 2003. Diffusionally accommodated interfacial sliding in metal-silicon systems. Acta Materialia 51, 2831-2846. Qu, J., Cherkaoui, M., 2007. Fundamentals of Micromechanics of Solidss. John Wiley & Sons, 25
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Inc. Shishvan, S.S., McMeeking, R.M., Pollock, T.M., Deshpande, V.S., 2017a. Discrete dislocation plasticity analysis of the effect of interfacial diffusion on the creep response of Ni single-crystal superalloys. Acta Materialia 135, 188-200. Shishvan, S.S., Pollock, T.M., Mcmeeking, R.M., Deshpande, V.S., 2017b. Interfacial diffusion in
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high-temperature deformation of composites: a discrete dislocation plasticity investigation.
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Sofronis, P., McMeeking, R., 1994. The effect of interface diffusion and slip on the creep resistance of particulate composite materials. Mechanics of materials 18, 55-68. Wang, Y.M., Weng, G.J., 1992. Transient Creep Strain of a Fiber-Reinforced Metal-Matrix Composite Under Transverse Loading. Journal of Engineering Materials & Technology 114,
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237-244.
Wakashima, K., Choi, B.H., Mori, T., 1990. Plastic incompatibility and its accommodation by
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diffusional flow: Modelling of steady state creep of a metal matrix composite. Materials Science & Engineering A 127, 57-64.
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Zheng, Q.S., Du, D.X., 2001. An explicit and universally applicable estimate for the effective properties of multiphase composites which accounts for inclusion distribution. Journal of the
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Mechanics & Physics of Solids 49, 2765-2788.
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Appendix A
For isotropic materials the elastic stiffness C is expressed in the following 6×6 matrix form.
2 2 2 C 0 0 0 0 0 0 0 0 0 26
0 0
0 0
0
0 0
0 0
0
0 0 0 , 0 0
(A1)
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where E / [2(1 )] and 2 / (1 2 ) . The Eshelby’s tensor for a long fiber (plane strain) is expressed as
A(4 1) 4 A A(5 4 ) 4 A 0 0 0 0
0 0
0 0 0 0
0 0
0 0 14 0 0 14 0 0
where A 1/ [8(1 )] .
x and y in Eq. (23) are expressed as
The coefficients
0 , 0 0 A(3 4 ) 0 0
(A2)
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A(5 4 ) A(4 1) 0 S 0 0 0
x E f f 1 E 2f f 1 m 1 5 m 4 m2 Em2 f 1 1 f 1 4 m 4 m2 1 m 2 m2 E f Em 1 m 2
2 2 1 4 m 3 f m 2 m 4 f m
f
m 4 m 3 f f 3 4 m 8
2 m
2 5
m
m 2 m2 /
2 2 2 2 Em f 1 f 2 f 1 E f f 1 m f 2 m 1 1 E f Em f 1 f 2 f m 4 f m 1 m / 2 E f 3 f 4 m 1 m Em f 1 1 f E f 1 2 E 1 f 1 4 f E f 1 1 4 /
2 E f E E f 2E E f
y
2 f
E 1 E 3 4 2 f f
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2 2 E f E 2f f m 1 1 m f 4 m 3 5 4 m Em2 f 1 1 f 3 m 2 m2 f m 4 m2 4 2 2 2 2 5 4 m 3 f m 2 m 4 f m f m 4 m 3 f 6 7 m 2 m f 8 m 8 m 1 E f Em f 1 1 m / 2 E f 3 f 4 m 1 m Em f 1 1 f / 2 2 2 2 Em f 1 f 2 f 1 E f f 1 m f 2 m 1 1 E f Em f 1 f 2 f m 4 f m 1 m 2 2 2 E f 1 E 3 f 4 f E f 1 5 4 / 2 E f E E f 2E f E f E 1 f E f 3 4 2
.(A3)
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Appendix B
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In inclusion-matrix type composites, the following relations hold for the average field
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quantities (Qu and Cherkaoui, 2007)
(1 f ) m f i , (1 f ) f m i
(B1)
where 0 and are the average stress and strain over the whole composite, 0 is the applied uniform stress, i ( m ) and ii (m ) are the average stress and strain of inclusions (matrix) and f is the inclusion volume fraction. The average stress and strain over matrix and inclusions satisfy
27
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m C m m . i Ci i
(B2)
Rewrite the second equation of Eq. (B1) to the following form
C10 (1 f )Cm1 m fCi1i .
(B3)
Eliminating m from the first equation of Eq. (B1) and Eq. (B3), we then get
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1 1 (Ci Cm1 )1 (C1 Cm1 )0 . f
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i
28
(B4)
A micromechanics method for transverse creep behavior induced by interface diffusion in unidirectional fiber-reinforced metal matrix composites Binbin Xu , Fenglin Guo PII: DOI: Reference:
S0020-7683(18)30388-3 https://doi.org/10.1016/j.ijsolstr.2018.09.024 SAS 10128
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
31 January 2018 2 August 2018 21 September 2018
Please cite this article as: Binbin Xu , Fenglin Guo , A micromechanics method for transverse creep behavior induced by interface diffusion in unidirectional fiber-reinforced metal matrix composites, International Journal of Solids and Structures (2018), doi: https://doi.org/10.1016/j.ijsolstr.2018.09.024
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Highlights
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Propose a new micromechanics method to analyze creep behavior induced by interface diffusion in metal matrix composites. Present an analytical solution for creep rate due to interfacial diffusion. Formulate a micromechanics model to estimate the stress field in the fiber accounting for interaction between fibers. Obtain overall creep strains and stress variation with time in fibers under constant external load by the incremental creep analysis procedures.
1
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A micromechanics method for transverse creep behavior induced by interface diffusion in unidirectional fiber-reinforced metal matrix composites
a
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Binbin Xua, Fenglin Guoa, b,*
School of Naval Architecture, Ocean and Civil Engineering (State Key Laboratory of Ocean Engineering), Shanghai Jiao Tong University, Shanghai 200240, China
b
Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240,
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China
Abstract
In this study, we present a new micromechanics method to predict the transverse creep rate
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induced by interface diffusion in unidirectional fiber-reinforced composites and evolution of overall creep strain under constant applied stress. An analytical solution for creep rate induced by
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interfacial diffusion, depending on the applied stress, fiber volume fraction and radius of the fiber, as well as the modulus ratio between the fiber and the matrix, is obtained. A micromechanics
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model is proposed to estimate the stress field in the fiber, which is related to the driving force for the interface diffusion, the normal stress on the interface. Comparison with finite element analysis
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shows that the present micromechanics model is of good accuracy, especially for high fiber volume fraction and large elastic modulus ratio between fiber and matrix. With the proposed
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micromechanics method based on the average field theory, the variation of overall creep strains and stresses with time in fibers under constant external load are analyzed by the incremental creep analysis procedures.
Keywords: Metal matrix composites (MMCs); Creep; Interface diffusion; Micromechanics
*
Corresponding author. E-mail: [email protected] (F. L. Guo). 2
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1. Introduction In recent years, particulate or fiber reinforced metal and intermetallic matrix composites (MMCs) have attracted considerable attention due to their good strength performance at high temperatures. The creep behavior of MMCs has been an essential topic of intense research. It is well-known that, in general, at modest temperatures the creep strength of metal and intermetallic matrix composites is better than that of the matrix alone because of the constraint of the
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reinforcement on the matrix. However, experimental results show that, at temperatures higher than about half of the melting temperature of the matrix, the composite strength is impaired and in some cases the strengthening imparted by the reinforcement is completely lost (Chun and Daniel, 1997; Nimmagadda and Sofronis, 1996). This creep behavior of composites is considered to be
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caused by diffusional relaxation and interface slip along the interface between the reinforcement and matrix. So far a great deal of effort has been devoted both theoretically and numerically in order to gain a better understanding on mechanisms of creep behavior of the composites. Chun and Daniel (1997) developed a micromechanics model based on the average field theory to predict
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transverse creep behavior in a unidirectional metal matrix composite. In their study, the matrix is considered as a creeping material following a Bailey-Norton power law, and creeping of the fibers
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and stress driven interfacial diffusion are neglected. Bullock et al. (1977) proposed an analytical model to predict creep rate of a Ni-Ni3AI-Cr3C2 eutectic composite by assuming that the fiber and
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matrix satisfy a power law of creep constitutive relationship, and have a homogeneous strain rate. McMeeking (1993) presented an asymptotic analysis for the power law creep of a matrix
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containing discontinuous rigid aligned fibers. Kim and McMeeking (1995) improved McMeeking’s model by incorporating the influences of interfacial slip and diffusion on the overall
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creep behavior of composites. Almost all analytical studies take the assumption of rigid reinforcements, which leads to inaccurate prediction of stress field along the interface and thus make it difficult to accurately assess the interface diffusion. In contrast with analytical investigations, numerical studies are capable of tackling general and complex problems, not only focusing on the overall creep behavior of composite based on the creep law of each constituent, but also paying attention to simulating the stress and strain relaxation processes, which bring in in-depth understanding to the mechanism of interfacial slip and diffusion induced creep behavior of the composites. Park and Holmes (1992) studied the 3
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tensile creep and creep-recovery behavior of fiber reinforced ceramic composites using finite element method with rectangular unit cells. The influences of interface condition, fiber volume fraction, as well as processing-related residual stresses were all considered (Park and Holmes, 1992). Later, Sofronis and McMeeking (1994) investigated the creep behavior of particulate composite materials by coupling the diffusion and slip along the interface with creep deformation of the matrix. It is found that normal stress gradients on the interface will induce diffusional flow
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along the interface accompanied by slip of the matrix over the reinforcement, which leads to the relaxation of the constraint. Finite element analysis on unit cell models showed that either diffusional relaxation or slip may knock down the creep resistance of the composite to levels even below the matrix strength (Sofronis and McMeeking, 1994). However, numerical studies are
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limited to fixed computational parameters and it is difficult to draw a broadly applicable conclusion based on the results of a small number of case studies. Thus, it is important to develop analytical solutions which are able to reveal the inherent characteristics of creep behavior caused by diffusional relaxation and interface slip.
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Experimental studies indicate that composites with micron- or nano-sized inclusions exhibit size effect on the overall effective elastic and plastic properties, which cannot be adequately
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captured by classical continuum mechanics theory. Discrete dislocation plasticity analysis has thus been conducted to investigate deformation of metal matrix composites and superalloys with
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micron- or nano-sized constituents at high temperatures (Shishvan et al., 2017a, 2017b; Shishvan et al., 2018). These studies show that elastic fields of dislocations in the matrix will affect the
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stress-driven interfacial diffusion in composites. In addition, the decrease in the strength of the composites due to interfacial diffusion is a result of a change in the dislocation structures within
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the matrix that, in turn, leads to a reduction in the matrix strength (Shishvan et al., 2017b). Metal matrix composites, in particular continuously reinforced MMCs, have demonstrated
potential for high temperature propulsion and airframe applications in aerospace industry because of their brilliant properties at elevated temperature in the fiber direction. Unfortunately, the transverse creep resistance and the creep-rupture life are much lower than those in the fiber direction due to the weak nature of the fiber/matrix interfacial bond and interfacial diffusion (Bednarcyk and Arnold, 2002; John et al., 1996). Therefore, a thorough investigation on the transverse creep performance is necessary in order to take full advantage of this type of metal 4
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matrix composites in engineering applications. Wang and Weng (1992) developed a local-field theory based on the combination of Mori-Tanaka method and the analytical solution of a three-phase cylindrically concentric solid to study the evolution of stress distribution in the matrix and overall creep strain of a fiber-reinforced metal-matrix composite. Their local-field theory had shown to provide a more accurate estimate for the overall transverse creep of the composite than the simpler Mori-Tanaka method. Chun and
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Daniel (1997) proposed a micromechanical model based on Mori-Tanaka average field theory to predict transverse creep deformation in a unidirectional metal matrix composite. Their results showed that the composite creep strain rate was less sensitive to the stress amplitude and temperature than that of the unreinforced matrix material. In order to elucidate the damage
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mechanisms under transverse creep loading of unidirectional metal matrix composites, Carrère et al. (2002) simulated the creep deformation process on a representative unit cell using the finite element method with coupling effect between matrix creep and interface damage taking into account. Numerical results showed that the global response of the composite subjected to a
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transverse creep loading strongly depends on the local interfacial behavior. Nevertheless, none of above mentioned researches on transverse creep of MMCs took the interfacial diffusion induced
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creep into consideration, which could be significant in high temperature applications. Recently, several researchers obtained the stress field in the fiber by Eshelby solution and
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concluded that the interface slip only has a short range effect compared to diffusional mass transport in unidirectional fiber reinforced composites (Li et al., 2014; Li and Li, 2012). They
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presented an analytical solution for the transverse creep rate contributed by interface diffusion. Their researches clearly visualized the roles of some important parameters for the interface creep,
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such as the fiber size and volume fraction. However, their method of calculating the normal stress gradient along the interface is only applicable for small fiber volume fraction. In unidirectional fiber reinforced composites, the creep behavior exhibits strong anisotropy
and creep rate is very low or negligible in the longitudinal direction but more pronounced in the transverse direction. Because the state of stress in the matrix is highly heterogeneous under transverse loading, it is quite difficult to obtain exact predictions of average creep behavior (Wang and Weng, 1992). A simple micromechanics model would be very desirable. In the present study, the normal stress gradient along the interface, which induces diffusional mass transport, is 5
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estimated by averaging results of self-consistent scheme (Hill, 1965) and the Mori-Tanaka method (Mori and Tanaka, 1973). Comparison of the proposed method with finite element analysis shows that the present method is able to give prediction of average stress in fiber with reasonable accuracy, especially for high fiber volume fraction and large contrast between fiber and matrix moduli. We also present an analytical solution of creep strain rate, which is formally similar to that of Coble creep in polycrystalline material (Coble, 1963) and creep rate due to diffusional mass
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transfer given by Mori et al. (1997). Then the deformation of fibers and creep behavior of metal matrix composites are well elucidated within the framework of the micromechanics method. With the proposed micromechanics method based on the average field theory, the accumulated creep strain and stress transfer occurring in fibers under constant external load are analyzed by the
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incremental creep analysis procedures. This study may bring new insights to understanding of transverse creep behavior and designing composite materials of higher creep resistance.
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2. Diffusional creep model
In numerical studies, the unit cell model is generally preferred if fiber arrangement in
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composites is assumed as a hexagonal or rectangular array (Berton et al., 2006; Mori et al., 1997; Nimmagadda and Sofronis, 1996). However, since this study is concerned with analytical analysis
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of transverse creep behavior of MMCs by micromechanics theory, it is not necessary to adopt the unit cell model in the present work. The fiber reinforcements are simulated as long parallel
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cylinders in the longitudinal direction. As depicted in Fig.1, a fiber with a radius of is embedded in matrix subjected to a general in-plane external stress σ 0 . The gray circle in Fig. 1
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stands for the cross-section of the long fiber. For simplicity, matrix and fibers are assumed to be homogeneous and isotropic and the interface between them is perfectly bonded. Upon loading the normal stress gradient along the interface causes matter flow from the sides to the poles of the fiber, resulting in creep deformation in the loading direction.
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σ0
y Vn
j
x
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matrix
σ0
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Fig. 1. A schematic diagram of interface diffusion in unidirectional fiber-reinforced composites.
The chemical potential of the atoms on the interface is (Li and Li, 2012)
0 s n ,
0 is the reference value of the potential, which is considered to be constant for a given
material.
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where
(1)
s and are the interface energy and the atomic volume respectively. is the
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curvature of the interface, a constant for a circular interface.
n is the normal stress acting on the
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interface, positive for tension and negative for compression. Besides, strain energy term has been taken to be negligible (Li and Li, 2012; Li et al., 2010; Sofronis and McMeeking, 1994).
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The volumetric flux per unit length crossing the interface is given by (Herring, 1950)
j
Di i Di i n , kT s kT s
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where Di is the interface diffusion coefficient,
(2)
i is an effective interface thickness, k is
the Boltzmann constant, T is the absolute temperature, and s is the arc length of the interface. The law of mass conservation along the interface requires
j Vn 0 , s
(3)
where Vn is the rate of accumulation or depletion of matter on the interface, positive for accumulation and negative for depletion. The above equation is the governing equation of the diffusional mass flow, which provides a constitutive type of relationship between stress and 7
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velocity on the interface boundary. Moving Vn to the right side of the equation, we have
Di i 2 n j Vn . s kT s 2
(4)
Thus, the creep rate of the fiber along the radial direction is given by
2Vn D 2 n i i . 2 kT s 2
(5)
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r
The crux of obtaining the diffusional creep rate is formulation of an appropriate micromechanics model to accurately estimate the stress distribution at the interface.
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3. Estimation of average stresses in fibers
Micromechanical analyses have been widely used to predict the effective properties (elasticity, plasticity, conductivity, etc.) of composites of the matrix-inclusion type. So far, a number of theoretical micromechanics methods have been proposed based on the pioneering work
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of Eshelby (Eshelby, 1957), including Mori-Tanaka method (Mori and Tanaka, 1973), self-consistent scheme (SCS) (Hill, 1965), differential scheme (DS) and double-inclusion model
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(Hori and Nemat-Nasser, 1993) as well as the interaction direct derivative estimate (IDD) (Zheng and Du, 2001), among others. Besides, there are two micromechanics models which possess
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mathematical rigor in their elasticity formulation and are applicable to high reinforcement volume fraction. They are composite spheres (cylinders) model and generalized self-consistent method
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(GSCM) (Christensen, 1998; Christensen and Lo, 1979). In this work, we are interested in how the average stress in inclusions could be estimated
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accurately by micromechanics methods. However, unlike the effective properties of composites, the stress in inclusions is much more difficult to estimate with decent accuracy, particularly for high fiber volume fraction and large elastic modulus ratio between fiber and matrix. It is generally recognized that Mori-Tanaka method tends to underestimate overall effective elastic moduli whereas self-consistent scheme tends to overestimate overall effective elastic moduli when the inclusion is stiffer than the matrix (Christensen, 1990). Results of finite element analysis in this study indicate that Mori-Tanaka method tends to underestimate the average stress in the loading 8
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direction in fibers while self-consistent scheme tends to overestimate the average stress in fibers for metal matrix composites. Accordingly, taking average of results of the Mori-Tanaka method and the self-consistent scheme may give a reasonable estimate for average stress in the fibers. The accuracy of this approach will be discussed later in this section for different fiber volume fractions and ratios of Young’s modulus between fiber and matrix. In Mori-Tanaka method, the ellipsoidal inclusion is placed within a homogenous matrix
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which had been subjected to the as-yet-unsolved average stress of the matrix before the inclusion was embedded. The average stress-strain relationship for the matrix is expressed as
σm σ0 σm Cm (ε0 εm ) ,
(6)
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where Cm is the stiffness tensor of the matrix, σ 0 is applied constant external stress, ε 0 is the elastic strain corresponding to the applied stress σ 0 in an equivalent body without reinforcement,
σm and ε m are respectively matrix stress and strain disturbances due to the presence of reinforcement. Similarly, with the help of Eshelby’s equivalent inclusion principle, the stress in the
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fiber can be expressed as
σ f σ0 σ f
(7)
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ED
C f Cm1 (σ 0 σ m ) ε f ε c , Cm Cm1 (σ 0 σ m ) ε f ε c ε*
where C f is the stiffness tensor of the fiber, σ f and ε f are respectively the disturbance
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stress and strain due to the presence of the inclusion and creep strain, εc is the creep strain in the
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inclusion and ε* is a fictitious eigenstrain which is introduced to relate the present problem to the equivalent inclusion problem. The relationship between the disturbance strains ε f and ε* is expressed as
ε f S(ε c ε* ) , where
(8)
S is the Eshelby’s tensor. The equilibrium condition with Eqs. (6) and (7) gives (1 f )σ m fσ f 0 ,
where f is the fiber volume fraction. 9
(9)
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Substituting Eq. (8) into Eq. (7) and making use of the equilibrium condition Eq. (9), the fictitious eigenstrain ε* can be solved as 1
ε* f (S I) S (C f Cm )1 Cm Cm1σ0 (1 f )(S I)εc ,
(10)
where I is the fourth-order symmetric identity tensor. From Eq. (7) and Eq. (9), the stress
σ f (1 f )Cm (S I)(ε* ε c ) σ m fCm (S I)(ε* ε c )
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disturbances in the fiber and matrix can be represented by ε* as
.
(11)
Substituting Eq. (11) into Eq. (7), the stress in the fiber can be expressed by
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σ f C f Cm1 (σ 0 σ m ) ε f ε c C f Cm1σ 0 f (S I)(ε* ε c ) S(ε* ε c ) ε c .
(12)
C f Cm1σ 0 (1 f )(S I)ε c [(1 f )S fI] *
Substituting Eq. (10) into this equation and after a long process of algebraic manipulation, the stress in the fiber can finally be given by
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σ Mf T C f ( A I ) 1 Cm1σ 0 (1 f )(S I)ε c ,
(13)
where A (1 f )S fI Cm (C f Cm ) . If there is no creep strain ( c 0 ) and volume
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1
fraction f 0 , Eq. (13) can be further reduced to the Eshelby solution 1
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σ Eshelby I Cm (I S)(Cf 1 Cm1 ) σ 0 . f
(14)
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On the other hand, in the self-consistent scheme, the applied loads and the interaction with other inhomogeneities are accounted for by assuming the inclusion is placed within a
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homogeneous matrix with as-yet-unknown effective properties, which had been subjected to uniform strain before the inclusion was embedded. It should be noted that the self-consistent scheme actually yields implicit equations for the effective properties. In order to obtain an explicit solution of the stress in the inclusion, we use Mori-Tanaka method to estimate the effective properties of the composite. The derivation process of stress expression is similar to the Eshelby solution, and the difference is that the inclusion is embedded in an effective medium of stiffness
C . Thus, the equivalent inclusion principle equation gives
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C f C1σ 0 ε ε c C C1σ 0 ε ε c ε*
(15)
where ε is the disturbance stress due to the presence of inclusions and creep strain, it can be expressed as
ε S(εc ε* )
(16)
Introducing Eq. (16) into Eq. (15), the eigenstrain ε* can be solved 1
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ε* C(S I) C f S (C f C)C1σ 0 C f εc εc
(17)
Once the eigenstrain ε* is known, the stress in the inclusion (fiber) can be computed, which is finally expressed in the following form after a series of mathematical operations
1
1
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σ SCS C f SC1 (C f C) I C1σ0 I (S I)1 SC1C f εc , f
(18)
where the effective stiffness tensor of the composite C is estimated through Mori-Tanaka method as
C (1 f )Cm fC f T (1 f )I fT
1
1
(19)
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with T I SCm 1 (C f Cm ) . From the preceding equation, the effective shear and
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in-plane bulk modulus of the fiber reinforced composite are
m 1
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f (1 f )(3 4 m ) / 4(1 m ) m / ( f m )
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f K K m 1 (1 f ) / 2(1 m ) K m / ( K f K m )
,
(20)
subscripts f
m denote the fiber and the matrix, respectively. The effective Young’s
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where , K and and
are the shear modulus, bulk modulus and Poisson’s ratio, respectively. The
modulus E and Poisson’s ratio v are related to shear and bulk moduli by
E (3
) K . 1 (1 ) 2 K
(21)
It is noted that the effective elastic modulus E and Poisson’s ratio obtained by Eq. (20) are the in-plane properties of the composite. Actually, a unidirectional metal matrix composite 11
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possesses transverse isotropy and the plane of isotropy is perpendicular to the direction of fiber. This property enables one to describe the in-plane properties of the composite by Young’s modulus and Poisson’s ratio. Taking the average of Eq. (13) and Eq. (18) gives
average f
1 M T ( f SCS ). f 2
(22)
stress components in the above equation can be given by
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This is the final expression of the estimated fiber stress. If the creep strain in the fiber is zero, the
x x 0 , y y 0 , xy 0 , where
(23)
0 is the applied far-field stress and expressions of coefficients x and y are
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presented in Appendix A for brevity.
In order to check the accuracy of Eq.(22), detailed FE-analyses are performed, in which the fiber packing of composites is approximated by a simple rectangular array (see Fig.2) since optical micrograph shows that the fiber packing in unidirectional fiber-reinforced composites can
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typically be idealized as either hexagonal or rectangular array (Park and Holmes, 1992). Considering the micromechanical analysis is meant to obtain effective properties of composites
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containing a large number of inclusions, the number of fibers in the FEM model should be large enough. When the overall size and fiber volume fraction of a finite element model are kept
PT
constant, the greater the number of fibers, the smaller the fiber size and vice versa. Accordingly, three different models are established, namely model (1), model (2) and model (3), in which the
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ratio of the cross-sectional area of a single fiber to the entire model is 1%, 5‰ and 1‰, respectively. These models are stressed in tension in the y-direction and appropriately meshed.
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According to Eshelby solution (Eshelby, 1957), the stress is uniform in a single elliptical inclusion embedded in an infinite matrix and the result has been proved to be mathematically rigorous. Although in reality the stress field in fibers is obviously heterogeneous, finite element results show that, if the influence of boundaries is ignored, the stress fluctuation in fibers is minimal in a wide range of fiber volume fraction, as shown in Fig. 2.
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y
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x
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Fig. 2. Finite element model (model (3)) of rectangularly arrayed fiber-reinforced composites with 28.9% fiber volume fraction consists of a total of 289 ( 17 17 ) fibers. The fluctuation of stress component in fibers in the loading direction y is around 7.4% ( m f 0.2 , 10 ).
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The accuracy of Eq. (22) is verified by FEA under various values of fiber volume fraction and modulus ratio of fibers to matrix. Figs. 3 and 4 present comparisons of average stress values in
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fibers calculated by FEA, the present micromechanics method, Eshelby solution, self-consistent scheme and Mori-Tanaka method. FEA results show that the relative size or the number of fibers
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has a slight effect on the average stress of fibers, which increases slightly with rising number of fibers when the fiber volume fraction is constant. This effect gradually decreases as the number of
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fibers increases and it is considered herein that the size of fibers in the model (3) is sufficiently small since the average stress variation of fibers is very small and neglected when the number of
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fibers is further increased. One can see that, when the modulus ratio between fibers and matrix is
E f Em 10 , the stresses in fibers calculated from Eq. (22) are very close to those of
FE-analysis for varying fiber volume fraction from 0% up to 50%, and the relative error with the results of FEM model (3) is less than 0.8%. As modulus ratio and fiber volume fraction increase, the accuracy of self-consistent scheme and Mori-Tanaka method will drop as well. Fig. 4 shows that even when the fiber volume fraction is 48.4% and modulus ration is up to 12, the estimate given by the present method is still in good agreement with that of FEA with very small error. 13
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Numerical results obviously show that the micromechanics model proposed in the present study is more accurate than direct use of Eshelby solution. Therefore, the average stress obtained by Eq. (22) gives a reasonable estimate of stresses in fibers.
1.35
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1.40
10
1.30
FEM-model (1) FEM-model (2) FEM-model (3) self-consistent scheme M-T method present method Eshelby solution
1.25
1.20
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Normalized stress y
1.45
1.15 0
10
20
30
40
50
Volume fraction (%)
Fig. 3. Comparison of normalized stress y 0 in the fiber calculated from Eq.(22), FE-analyses, Eshelby
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solution, self-consistent scheme and Mori-Tanaka method for varying fiber volume fraction ( m f 0.2 ,
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10 ).
1.6
Normalized stress y/
AC
CE
PT
1.4 1.2 1.0 0.8
FEM-model (3) self-consistent scheme Eshelby solution M-T method present method
0.6
f=0.484 0.4 0.2 0.0 0
2
4
6
8
10
12
Elastic modulus ratio
Fig. 4. Comparison of normalized stress y 0 in the fiber calculated from Eq.(22), FE-analyses, Eshelby solution, self-consistent scheme and Mori-Tanaka method for varying
f 48.4% ). 14
in a range of 0–12 ( m f 0.2 ,
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To further examine the accuracy of Eq. (22) we write average stress in the fiber in the following weighted average form
f (1 w) Mf T w SCS , f
(24)
where the weight w is a function of the fiber volume fraction f and the elastic modulus ratio
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between fibers and matrix. It can be determined by making the value of f in Eq. (24) fit the FEA results. An approximate formula of polynomial fitting for the weight w is given below
w( f , ) (1.0200305 4.2943 f 16.6368 f 2 15.8651 f 3 )
(1.08453 0.15671 0.0132665 2 0.00040065 3 )
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where f [0.064, 0.484] and [5,12] .
(25)
For a typical value 10 , the weight factor w changes from 0.445 to 0.546 when the fiber volume fraction varies from 6.4% to 48% . It can be seen from Fig. 3 that both Mori-Tanaka method and self-consistent scheme will approach the Eshelby solution for the case of
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small fiber volume fraction.
By micromechanics theory, the average stress in inclusions or fibers is related to the overall
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effective stiffness tensor of the composite by
1 1 (Ci Cm1 )1 (C1 Cm1 )σ 0 . f
(26)
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σi
Derivation of Eq. (26) is given in Appendix B. As mentioned earlier in this section, Mori-Tanaka
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method tends to underestimate overall effective elastic moduli whereas self-consistent scheme tends to overestimate overall effective elastic moduli when the inclusion is stiffer than the matrix.
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Therefore, from Eq. (26) we deduce that the actual average stress in the fiber lies between 1 f
(Ci1 Cm1 )1 (CM1T Cm1 )σ0 and
1 f
1 (Ci1 Cm1 )1 (CSCS Cm1 )σ0 for metal matrix composites.
Based on these arguments, we conclude that taking average of results of the Mori-Tanaka method and the self-consistent scheme provides a simple and reasonable approach to estimate average stress in inclusions or fibers in view of the complexity of fiber distribution and interaction in MMCs. 4. Transverse creep rate 15
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As mentioned before, in the fiber-reinforced composite with a rectangular array of fiber packing, the stresses in fibers are approximately uniform and can be estimated analytically even though the stress distribution in matrix is complicated. Thus, stress components in fibers in a polar coordinate system can be expressed by
x y x y cos 2 2 2 y x y x cos 2 , 2 2 y r x sin 2 2 r
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where
(27)
x and y are stress components in the directions x and
y as presented in Eq.(23),
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which are also the principal stresses in fibers for this problem. r r is just the normal stress n along the interface between the fiber and the matrix.
From Eqs. (27) and (5) and noting that ds d , we have
2 Di i ( x y ) cos(2 ) 3kT . 2 Di i 3 x y 0 cos(2 ) kT
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r
(28)
Then, it follows from Eq. (28) that
2 Di i x y 0 3kT . 2 Di i 3 x y 0 kT
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x r 0
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y r 90
(29)
It is obvious that the creep strain rate in the 45 direction is zero by Eq. (28), which implies
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xy =0. The average creep strain rate of the fiber can be expressed in the following form
where
x ε c 0 0
0
y 0
0 0 , 0
(30)
x and y are components of creep strain rate in directions of x-axis and y-axis,
respectively. Eqs. (28) and (29) indicate that the creep rate of the fiber depends on the fiber volume 16
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fraction, the radius of the fiber, and material properties of the fiber and matrix as well as the applied stress. It should be noted that we have taken account of interaction between fibers in the process of deriving creep rate expression. As a result, the expression of creep rate obtained in the present study is expected to be valid for a wide range of fiber volume fraction up to 50%. Although the creep rate component
y is formally similar to those of previous works, we also
CR IP T
note that the creep rate in Eq. (30) is presented in the form of tensor and is essentially different from the previous works (Coble, 1963; Li and Li, 2012; Wakashima et al., 1990), which only consider the creep rate in the loading direction. However, all these results predict linear dependence of creep rate on the applied stress.
So far, we have derived the tensor form of creep rate. As can be seen from Eq. (28), the creep
r is anti-symmetrical with respect to the direction 45
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rate component
and
r (45 ) 0 , which means there is no creep occurring in the fiber along the diameter in the 45 direction and the diffusional creep deformation in fibers is incompressible. The illustration
y
CE
PT
ED
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of fiber deformation caused by creep due to interfacial diffusion is schematically shown in Fig. 5.
45°
AC
x -45°
Fig. 5. Illustration of stress-directed diffusional mass transfer along the interface between fiber and matrix in the 17
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composite
5. Transverse creep behavior under constant stress Metal-matrix composites (MMCs) have found many applications in aircraft and automotive industries as well as civil engineering, as structural components due to their excellent physical and mechanical properties at high temperatures. Creep behavior is usually a major concern when
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evaluating safety of structural components made of metal-matrix composites operating at high temperatures or high stress level. For example, creep of a turbine blade will cause the blade to contact the casing, resulting in failure of the blade. Therefore, the creep of MMCs should be precisely analyzed to guarantee structural safety. When an external constant stress is applied, the
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fiber and matrix are subjected to an initial stress distribution upon equilibrium. The corresponding initial stress gradient along the interface causes diffusional mass transport, which is generally assumed as plastic strain, in turn, disturbs the initial stress distribution. In this study, we investigate the creep behavior due to interfacial diffusion in the composite, and the fiber and
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matrix are assumed to deform elastically due to the applied stress and the interfacial diffusion. Eventually, the transverse creep behavior can be predicted by incremental analysis procedures.
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The total average strain of the composite can be obtained from Eqs. (22) and (30) t
ε(t ) (1 f )Cm1σ m (t ) fCf 1σ f (t ) f εc dt ,
(31)
t0
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where t is time and σ m (t ) satisfies (1 f )σ m (t ) fσ f (t ) σ0 . In the process of creep
CE
deformation, the creep strain due to the diffusional mass transfer at the inclusion/matrix interface, which can be introduced into the fiber as eigenstrain, may affect the stress distribution in the fiber
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and matrix. Although it is hard to obtain explicit solution for this problem, we can calculate the overall average strain and the average stress in the fiber at any time t by incremental analysis procedures as long as the initial creep rate is known. At initial moment t t0 , no creep occurs in the composite. The stress fields in the fiber and
matrix are respectively
1 1 σ f (t0 ) C f ( A I) 1 Cm1 SC1 (C f C) I C1 σ 0 . 2 σ m (t0 ) [σ 0 fσ f (t0 )] / (1 f )
18
(32)
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For a small time increment dt , we assume that the creep rate is constant and the creep strain induced by interface diffusion can be expressed as
c (t )dt , then the stress fields in matrix and
fiber become
1 C (1 f )(S I )(A I ) 2
σ f (t dt )
1 1 C f ( A I) 1 Cm1 SC1 (C f C) I C1 σ 0 2 1
f
I (S I )1 SC1C f
t ε c dt ε c (t )dt t 0
(33)
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and
1
σ m (t dt ) [σ0 fσ f (t dt )] / (1 f ) . The updated total strain in the composite is
(34)
(35)
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t ε(t dt ) (1 f )Cm1σ m (t dt ) fCf 1σ f (t dt ) f εc dt εc (t )dt . t 0
These incremental calculations can be carried out with the help of mathematical software like Mathematica or Matlab.
In order to examine the creep behavior of composites induced by interfacial diffusion, the
(t ) / (t0 ) and (t ) / (t0 ) , is plotted as a function of normalized time
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direction,
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normalized transverse creep strain and the creep strain rate of the composite in the loading
t 2Diit / 3kT for different fiber volume fraction f and elastic modulus ratio in
PT
Figs. 6 and7. The applied external stress is 20Mpa and the elastic modulus of matrix is 50Gpa. The
AC
CE
Poisson’s ratios of fiber and matrix are both 0.2.
2.8 2.6 2.4
1.0
=10
0.1 =0.2 =0.3 =0.4
0.8
2.2 0.6
=0.1 =0.2 =0.3 =0.4
ε(t ) 2.0 ε(t 0 ) 1.8
ε( t ) ε( t0 ) 0.4
1.6 1.4
0.2
1.2 1.0 0
5
10
15
20
t (0.0025)
19
25
30
35
0.0 40
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Fig. 6. The normalized creep strain (solid lines) and the corresponding strain rate (dashed lines) as a function of the normalized time t 2Di i t / 3 kT for composites with different fiber volume fractions at fixed 10 and f m 0.2 .
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2.0
1.0
=0.3
=2 =5 =10 =12
2.2
0.8
1.8
0.6
=2 =5 =10 =12
ε( t ) 1.6 ε( t 0 ) 1.4
ε( t ) ε( t0 )
0.4
0.2
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1.2
1.0 0
5
10
15
20
25
30
35
0.0 40
t (0.0025)
Fig. 7. The normalized creep strain (solid lines) and the corresponding strain rate (dashed lines) as a function of
M
the normalized time t 2Di i t / 3 kT for composites with different elastic modulus ratios at fixed f 0.3
ED
and f m 0.2 .
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As can be seen from Figs. 6 and 7, both fiber volume fraction and elastic modulus of fiber have a large impact on creep resistance of the composite material. As far as the creep due to
CE
interfacial diffusion is concerned, higher volume fraction or elastic modulus of the fiber results in larger total creep strain of the composite, although this favors the overall elastic properties. It is
AC
observed that fiber volume fraction has a bigger effect on creep strain than the elastic modulus of fiber. This is because larger fiber volume fraction means that the composite contains larger area of interface between the fiber and the matrix, leading to more interfacial diffusion. The maximum creep strain in the loading direction, which occurs in the case 10 and f 0.4 , is over 2.6 times the initial elastic strain. Besides, these results show that the creep due to interfacial diffusion will mainly contribute to the primary creep regime as the corresponding creep strain rate eventually approaches zero. The change in the stress field of the composite is another major concern. Since stress field in the matrix is highly heterogeneous, only the stresses in the fiber, 20
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where the gradual stress transfer occurs, are analyzed. The two perpendicular stress components in the fiber are shown in Fig. 8 and Fig. 9 in various fiber volume fractions and elastic modulus ratios. Figs. 8 and 9 show that the stress component the elapse of time while the stress component
y in the loading direction decreases with
x changes just in the opposite way. This
phenomenon indicates that the normal stress gradient and shear stress along the interface will
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decrease gradually and explains the reason why the creep rate induced by interfacial diffusion eventually drops to zero. It is also found that the stress in fiber is more sensitive to the elastic modulus ratio than the fiber volume fraction. These results may bring new insights to
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understanding of diffusional creep behavior.
1.4
Stress component y
1.2 1.0
σ0
0.8
M
σ f (t )
0.6 0.4
Stress component x
ED
0.2 0.0
5
10
15
20
25
30
35
40
t (0.0025)
PT
0
CE
Fig. 8.The normalized stress in fibers as a function of the normalized time t 2Di i t / 3 kT for the
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composites with different fiber volume fractions at fixed 10 and f m 0.2 .
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1.4
Stress component y
1.2
1.0 0.8
σ f (t ) 0.6
Stress component x
0.4 0.2 0.0 0
5
10
15
20
t (0.0025)
25
30
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σ0
35
40
Fig. 9. The normalized stress in fibers as a function of the normalized time t 2Di i t /
6. Discussions and concluding remarks
f m 0.2 .
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composites with different Young’s module ratios at fixed f 0.3 and
3 kT for the
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In this study, we have developed a new micromechanics method to predict the transverse creep rate induced by interface diffusion in unidirectional fiber-reinforced composites and
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evolution of overall creep strain under constant applied stress. The driving force for the interface diffusion is the normal stress acting on the interface, which can be estimated by micromechanics
PT
methods. The proposed micromechanics model, which takes the average of the results of self-consistent scheme and Mori-Tanaka method, enables one to give estimate of the stress field in
CE
the fiber of fiber reinforced metal matrix composites with reasonable accuracy. Comparison with finite element analysis shows that the present method is better than direct use of Eshelby solution,
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especially for high fiber volume fraction and large contrast between fiber and matrix elastic moduli. For the case of small fiber volume fraction (below 10%), results of Eshelby solution, Mori-Tanaka method and self-consistent scheme are close to each other. When the fiber volume fraction is larger than 20%, the difference among these methods is getting obvious. For composites with small fiber volume fraction, we still can use Eshelby solution to quantify major features of interfacial diffusion creep in metal matrix composites (Li and Li, 2012). 2012) For composites with large fiber volume fraction, the micromechanics method proposed in this study 22
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offers a better choice for investigation of interfacial diffusion creep since it is able to give more accurate estimate of stress in fibers. An analytical solution of creep rate induced by interfacial diffusion, depending on the applied stress, volume fraction and radius of the fiber, as well as the modulus ratio between the fiber and the matrix, is presented. The deformation of fiber and composite as well as evolution of overall creep strain can be well described by this micromechanics method. With the proposed
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micromechanics method based on the average field theory, the creep strains and stress transfer under constant external force occurring in fibers are analyzed by the incremental creep analysis procedures. The methodology and results of this study may bring new insights to understanding of
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the creep behavior and designing composite materials of higher creep resistance.
The interfacial diffusion and interfacial slip are generally regarded to be related to each other. Petersonet al. (2003) proposed that interfaces may slide via interface diffusion controlled diffusional creep, which had been validated experimentally. On the other hand, interface slip may
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make stress field along the interface redistribute, and in turn, affect interface diffusion driven by the normal stress along the interface. Li Y and Li Z (2012) mentioned that the interface slip is
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limited to the local regions of the interface and only has a short range effect. Besides, the effect of interface slip can be taken into account by incorporating it into the creep of matrix. Therefore, the
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present micromechanics method without taking interface slip into consideration is applicable when we only focus on the creep due to interfacial diffusion.
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In this work, we only studied the creep due to mass diffusion at interface between the fiber and matrix and did not consider the creep of matrix. Evidently, the creep induced by the interfacial
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diffusion is always coupled with matrix creep in the real metal matrix composite materials. In general, the matrix creep could make stress in the composite redistribute, and will, in turn, affect the interfacial diffusion. On the other hand, the development of creep strain in fibers due to interface diffusion will counteract the applied stress and thus make the average stress in the matrix increase. Only after the matrix creep and the interface creep are thoroughly investigated individually, would it be possible to fully understand the intricate interaction between them. This is beyond the scope of the present study and will be the task for future research.
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Acknowledgement
The authors gratefully acknowledge the financial support by the National Natural Science Foundation of China through Grant No. 11272206.
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References
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Appendix A
For isotropic materials the elastic stiffness C is expressed in the following 6×6 matrix form.
2 2 2 C 0 0 0 0 0 0 0 0 0 26
0 0
0 0
0
0 0
0 0
0
0 0 0 , 0 0
(A1)
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where E / [2(1 )] and 2 / (1 2 ) . The Eshelby’s tensor for a long fiber (plane strain) is expressed as
A(4 1) 4 A A(5 4 ) 4 A 0 0 0 0
0 0
0 0 0 0
0 0
0 0 14 0 0 14 0 0
where A 1/ [8(1 )] .
x and y in Eq. (23) are expressed as
The coefficients
0 , 0 0 A(3 4 ) 0 0
(A2)
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A(5 4 ) A(4 1) 0 S 0 0 0
x E f f 1 E 2f f 1 m 1 5 m 4 m2 Em2 f 1 1 f 1 4 m 4 m2 1 m 2 m2 E f Em 1 m 2
2 2 1 4 m 3 f m 2 m 4 f m
f
m 4 m 3 f f 3 4 m 8
2 m
2 5
m
m 2 m2 /
2 2 2 2 Em f 1 f 2 f 1 E f f 1 m f 2 m 1 1 E f Em f 1 f 2 f m 4 f m 1 m / 2 E f 3 f 4 m 1 m Em f 1 1 f E f 1 2 E 1 f 1 4 f E f 1 1 4 /
2 E f E E f 2E E f
y
2 f
E 1 E 3 4 2 f f
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2 2 E f E 2f f m 1 1 m f 4 m 3 5 4 m Em2 f 1 1 f 3 m 2 m2 f m 4 m2 4 2 2 2 2 5 4 m 3 f m 2 m 4 f m f m 4 m 3 f 6 7 m 2 m f 8 m 8 m 1 E f Em f 1 1 m / 2 E f 3 f 4 m 1 m Em f 1 1 f / 2 2 2 2 Em f 1 f 2 f 1 E f f 1 m f 2 m 1 1 E f Em f 1 f 2 f m 4 f m 1 m 2 2 2 E f 1 E 3 f 4 f E f 1 5 4 / 2 E f E E f 2E f E f E 1 f E f 3 4 2
.(A3)
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Appendix B
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In inclusion-matrix type composites, the following relations hold for the average field
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quantities (Qu and Cherkaoui, 2007)
(1 f ) m f i , (1 f ) f m i
(B1)
where 0 and are the average stress and strain over the whole composite, 0 is the applied uniform stress, i ( m ) and ii (m ) are the average stress and strain of inclusions (matrix) and f is the inclusion volume fraction. The average stress and strain over matrix and inclusions satisfy
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m C m m . i Ci i
(B2)
Rewrite the second equation of Eq. (B1) to the following form
C10 (1 f )Cm1 m fCi1i .
(B3)
Eliminating m from the first equation of Eq. (B1) and Eq. (B3), we then get
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1 1 (Ci Cm1 )1 (C1 Cm1 )0 . f
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i
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(B4)