Q. Tai. A. Mocellin/Ceramics International 25(1999)393-408 and p and n are constants termed the inverse grain size and exponent and the stress exponent, respectively. The dif- fusion coefficient D may be expressed as Do exp(-Q/ RD, where Do is a frequency factor, Q is the apparent PI=P2 PI activation energy, and R is the gas constant Vi is the volume fraction of phase i; ni is the viscosity For two phase composites, there are several equations undergoing Newtonian viscous flow which may express or predict their high temperature n= Vinl +v2772: qi is the phase'stress-concentration reep behaviours factor, Vig1+ V292=1; pi is internal stress caused by the In composites, where the second phase can be mismatch in creep strains between the phases sidered rigid, Raj and Ashby model [1] assumes that the VIPI+V2P2=0. hard second phase particles in the grain boundary of The Eqn. (5)is also valid for the case wherein one of matrix limit the grain boundary sliding and gives he phases is nondeformable by creep if diffusional mass transport around the purely elastic phase is taken into account R 2. 2. Theoretical models for plastic deformation where V is the second phase volume content, r is second phase grain radius, q and n are phenomenological There are several theoretical models for creep defor exponents and C is a constant mation. In general, they can be divided into two broad Chen model [2] considers the composites as a model categories: boundary mechanisms [5-ll and lattice system of a soft matrix containing equiaxed and rigid mechanisms [5, 12]. Boundary mechanisms rely on the inclusions. Based on a phenomenological constitutive presence of grain boundaries and occur only in poly- equation and a second phase continuum mechanics crystalline materials. They are associated with some model, his model gives: dependence on grain size so that p> l. Lattice mechan- isms are independent of the presence of grain bound E=(1-V)2+n2 () aries and occur both in single crystal and polycrystalline materials. They occur within the grain interiors and are where V is the second phase volume content, Eo is the independent of grain size, so p=0 strain rate of the reference matrix, n is the stress expo The boundary mechanisms can be subdivided into nent of the matrix four categories: diffusion creep, [5-7] interface reaction Ravichandran and Seetharaman model [3] considers controlled diffusion creep [8], grain boundary sliding that a rigid and noncreeping second phase distributes and grain rearrangement [5,6,9, 101, and cavitation creep uniformly in a continuous creeping matrix, and they and microcracking [11]. In diffusion creep where vacan- develop a simple continuum mechanics model to predict cies may flow from the zones experiencing tension to the steady state creep rates of composites those in compression either through the crystalline lat tice(Nabarro-Herring creep) or along the grain (1+C2 boundaries( Coble creep), the individual grains become (1+C) (4) elongated along the tensile axis Fig. 1). When grain +(1+C boundaries do not act as perfect sources or sinks for vacancies, the process of creating or annihilating point where C=l-I, v is the second phase volume content, n is the stress exponent of the matrix, A is the constant of the matrix For a two phase composite in which each phase undergoes diffusional creep, Wakashima and Liu give a viscoelastic constitutive equation corresponding to SUND/AR pring- dashpot model [4: A1-p(-分) where △EV1 Fig. 1. Diffusion flow by lattice(Nabarro-Herring creep)or by grain boundaries( Coble creep)and p and n are constants termed the inverse grain size exponent and the stress exponent, respectively. The dif￾fusion coecient D may be expressed as Do exp (ÿQ/ RT), where Do is a frequency factor, Q is the apparent activation energy, and R is the gas constant. For two phase composites, there are several equations which may express or predict their high temperature creep behaviours. In composites, where the second phase can be con￾sidered rigid, Raj and Ashby model [1] assumes that the hard second phase particles in the grain boundary of matrix limit the grain boundary sliding and gives: "_ ˆ C n dprqV exp ÿ Q RT   …2† where V is the second phase volume content, r is second phase grain radius, q and n are phenomenological exponents and C is a constant. Chen model [2] considers the composites as a model system of a soft matrix containing equiaxed and rigid inclusions. Based on a phenomenological constitutive equation and a second phase continuum mechanics model, his model gives: "_ ˆ "_o…1 ÿ V† 2‡n=2 …3† where V is the second phase volume content, "_o is the strain rate of the reference matrix, n is the stress expo￾nent of the matrix. Ravichandran and Seetharaman model [3] considers that a rigid and noncreeping second phase distributes uniformly in a continuous creeping matrix, and they develop a simple continuum mechanics model to predict the steady state creep rates of composites: "_ ˆ A …1 ‡ C† 2 …1‡C† 1=n C   ‡ …1 ‡ C† 2 ÿ 1 2 4 3 5 n …4† where C ˆ 1 V 1=3 ÿ1, V is the second phase volume content, n is the stress exponent of the matrix, A is the constant of the matrix. For a two phase composite in which each phase undergoes di€usional creep, Wakashima and Liu give a viscoelastic constitutive equation corresponding to a spring-dashpot model [4]: " ˆ
E Eu 1 ÿ exp ÿ t  n o   ‡ 1    …5† where
E Eu  V1 p1 q1 ÿ 1   2 ˆ V2 p2 q2 ÿ 2 u  2 and   1 p1 ÿ p2 ˆ 1 p1 1 ÿ V1 1    ˆ ÿ 2 p2 1 ÿ V2 2    Vi is the volume fraction of phase i; i is the viscosity of phase i undergoing Newtonian viscous ¯ow,   V11 ‡ V22; qi is the phase `stress-concentration' factor, V1q1+V2q2=1; pi is internal stress caused by the mismatch in creep strains between the phases, V1p1 ‡ V2p2 ˆ 0. The Eqn. (5) is also valid for the case wherein one of the phases is nondeformable by creep if di€usional mass transport around the purely elastic phase is taken into account. 2.2. Theoretical models for plastic deformation There are several theoretical models for creep defor￾mation. In general, they can be divided into two broad categories: boundary mechanisms [5±11] and lattice mechanisms [5,12]. Boundary mechanisms rely on the presence of grain boundaries and occur only in poly￾crystalline materials. They are associated with some dependence on grain size so that p51. Lattice mechan￾isms are independent of the presence of grain bound￾aries and occur both in single crystal and polycrystalline materials. They occur within the grain interiors and are independent of grain size, so p=0. The boundary mechanisms can be subdivided into four categories: di€usion creep, [5±7] interface reaction controlled di€usion creep [8], grain boundary sliding and grain rearrangement [5,6,9,10], and cavitation creep and microcracking [11]. In di€usion creep where vacan￾cies may ¯ow from the zones experiencing tension to those in compression either through the crystalline lat￾tice (Nabarro±Herring creep) or along the grain boundaries (Coble creep), the individual grains become elongated along the tensile axis Fig. 1). When grain boundaries do not act as perfect sources or sinks for vacancies, the process of creating or annihilating point Fig. 1. Di€usion ¯ow by lattice (Nabarro±Herring creep) or by grain boundaries (Coble creep). 396 Q. Tai. A. Mocellin / Ceramics International 25 (1999) 395±408