《复合材料 Composites》课程教学资源（学习资料）第五章陶瓷基复合材料_Finite element analysis of the fatigue.pdf_大学文库

Finite element analysis of the fatigue crack growth rate in transformation toughening ceramics P. Vena *, D. Gastaldi, R. Contro, L. Petrini Dipartimento di Ingegneria Strutturale, Laboratory of Biological Structure Mechanics, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy Received 15 January 2005 Available online 15 August 2005 Abstract The fatigue crack growth rate in the zirconia tetragonal polycrystal is analyzed through the finite element method. In order to achieve this purpose, a continuum based constitutive law for materials subjected to phase transformations has been suitably implemented into a commercial finite element code. The fatigue crack growth in a notched beam, subjected to a cyclic four points bending load, has been investigated through a sensitivity analyses with respect to the two most relevant constitutive parameters: one accounting for the amount of the transformation strain and one accounting for the activation energy threshold. The fatigue crack growth rate typical of transforming materials is characterized by two distinct stages: at the beginning of the crack propagation process, the crack growth rate exhibits a negative dependency on the applied stress intensity factor; thereafter, a linear positive dependency is observed. This two stage process is well caught by the finite element model presented in this paper. Moreover, the response of the computational analyses has shown that the strength of the transformation process is determinant for the crack growth process to be arrested. 2005 Elsevier Ltd. All rights reserved. Keywords: Transformation toughening; Fatigue; Zirconia; Stress intensity factor; Finite element method 0749-6419/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2005.05.007 * Corresponding author. Tel.: +39 02 2399 4236; fax: +39 02 2399 4220. E-mail address: vena@stru.polimi.it (P. Vena). www.elsevier.com/locate/ijplas International Journal of Plasticity 22 (2006) 895–920

P. Vena et al. International Journal of Plasticity 22(2006)895-920 1. Introduction Strength and fatigue life of the ceramic materials are extremely xisting flaws which develop during the manufacturing process. These flaws act as initial defects determining stress concentration and potential crack growth. This owed to the brittle nature of ceramics that makes the issue of fracture toughness of major concern in designing reliable ceramic components to be used in struc- tural applications. Nowadays, advanced ceramics offer new possibilities in design ing structural components thanks to their toughening mechanisms, based on the phase transformations phenomena, which delay the crack propagation. The trans- formation toughening mechanisms are peculiar features of partially stabilized zir conia (PSZ), tetragonal zirconia polycrystal (TZP)and zirconia toughened alumina (ZTA), and can be different according to the chemical composition of the material Pure zirconia is an oxide(ZrO2) which cannot be obtained at room tempera ture without being damaged by distributed microcracks due to phase transforma tions occurring during the cooling process. For this reason pure zirconia is not rocess, by means of doping with calcia(CaO), magnesia(MgO), yttri (Y203) or ceria(CeO2) can provide a material which does not exhibit damage or smeared cracks at room temperature. The material considered in this paper is an yttria doped tetragonal zirconia polycrystal (TZP). From the chemical microstructural point of view, this material is constituted of tetragonal crystals (t-ZrO2) only. The peculiar mechanical property of the zirconia-based ceramics, and particularly of the TzP, is the phase transformation induced by the stress state. The tetragonal phase can spontaneously undergo to phase transformation at room temperature to a more stable monoclinic phase if a sufficient amount of mechanical energy(stress state)is provided to the system. The phase transfor- mation is accompanied by an irreversible transformation strain state. The amount of this transformation strain depends on the lattice parameters misfit between the parent and the generated phases and on several other parameters like crystal orientations It is well known that the fracture toughness of zirconia-based ceramics can be greatly enhanced by exploiting the stress-induced phase transformations in the zirco- nia particles or zirconia grains(Evans and Heuer, 1980; Munz and Fett, 1998; Hann ink et al., 2000; Rauchs et al., 2001, 2002; Kelly and Rose, 2002). This phenomenon produces the so-called frontal toughening mechanism or extrinsic shielding mechanism which originates from the development of permanent or transformation strains, such as those produced by the martensitic-type phase transformations, in the crack wake (Ritchie, 1999) When the crack propagates, the crack's tip travels and the stress concentra- tion migrates as well. This promotes new phase transformations in ahead of the crack tip; whereas in the wake of the crack the particles or grains previously transformed stay in their new transformed state. An area of material subjected to permanent strains(transformation strains) is therefore created along

1. Introduction Strength and fatigue life of the ceramic materials are extremely sensitive to preexisting flaws which develop during the manufacturing process. These flaws act as initial defects determining stress concentration and potential crack growth. This is owed to the brittle nature of ceramics that makes the issue of fracture toughness of major concern in designing reliable ceramic components to be used in structural applications. Nowadays, advanced ceramics offer new possibilities in designing structural components thanks to their toughening mechanisms, based on the phase transformations phenomena, which delay the crack propagation. The transformation toughening mechanisms are peculiar features of partially stabilized zirconia (PSZ), tetragonal zirconia polycrystal (TZP) and zirconia toughened alumina (ZTA), and can be different according to the chemical composition of the material. Pure zirconia is an oxide (ZrO2) which cannot be obtained at room temperature without being damaged by distributed microcracks due to phase transformations occurring during the cooling process. For this reason pure zirconia is not used as a structural material. A stabilization of the cubic phase during the cooling process, by means of doping with calcia (CaO), magnesia (MgO), yttria (Y2O3) or ceria (CeO2) can provide a material which does not exhibit damage or smeared cracks at room temperature. The material considered in this paper is an yttria doped tetragonal zirconia polycrystal (TZP). From the chemical microstructural point of view, this material is constituted of tetragonal crystals (t-ZrO2) only. The peculiar mechanical property of the zirconia-based ceramics, and particularly of the TZP, is the phase transformation induced by the stress state. The tetragonal phase can spontaneously undergo to phase transformation at room temperature to a more stable monoclinic phase if a sufficient amount of mechanical energy (stress state) is provided to the system. The phase transformation is accompanied by an irreversible transformation strain state. The amount of this transformation strain depends on the lattice parameters misfit between the parent and the generated phases and on several other parameters like crystal orientations. It is well known that the fracture toughness of zirconia-based ceramics can be greatly enhanced by exploiting the stress-induced phase transformations in the zirconia particles or zirconia grains (Evans and Heuer, 1980; Munz and Fett, 1998; Hannink et al., 2000; Rauchs et al., 2001, 2002; Kelly and Rose, 2002). This phenomenon produces the so-called frontal toughening mechanism or extrinsic shielding mechanism which originates from the development of permanent or transformation strains, such as those produced by the martensitic-type phase transformations, in the crack wake (Ritchie, 1999). When the crack propagates, the cracks tip travels and the stress concentration migrates as well. This promotes new phase transformations in the zone ahead of the crack tip; whereas in the wake of the crack the particles or grains previously transformed stay in their new transformed state. An area of material subjected to permanent strains (transformation strains) is therefore created along 896 P. Vena et al. / International Journal of Plasticity 22 (2006) 895–920

P. Vena et al. International Journal of Plasticity 22(2006)895-920 the path of the crack. If this strain field is permanent, i.e., in the case in which the transformation phenomenon is irreversible, a beneficial effect on the fracture toughness is obtained through a reduction of the stress intensity factor. Thi means that, for a given applied load which would produce an applied stress intensity factor in a non-transforming material, it can be defined an effective stress intensity factor that accounts for the effects given by the transformation strains in transforming materials. The transformation strains in the wake of the crack modifies the stress state around the crack tip in a such a way that the effective stress intensity factor is smaller than the applied stress intensity Moreover, it has been widely proved through experimental tests that the frontal toughening mechanism based on the tetragonal to monoclinic transformation occur- ring in the Ziconia ceramics gives the material a peculiar cyclic fatigue behaviour especially for small cracks(Steffen et al., 1991) The phenomenon may be described as a two stages process: at the beginning of the fatigue crack propagation process, the crack growth rate exhibits a nega tive dependency on the applied stress intensity factor; thereafter, after having reached a minimum, a linear positive dependency has been found. In the first stage, the relationship between the applied stress intensity factor and the crack growth rate is strongly dependent on the strength of the transformation capabil- ity of the material; whereas, in the second stage the relationship between the ap- which is non-dependent on the transformation strength. This latter is the typical behaviour of long cracks that follows the Paris law like the fatigue crack growth in metals From the results of the numerical experiments and the comparison with experi- mental finding presented in Karihaloo and Andreasen(1996), it would appear that the power-law is valid for both small as well as long cracks; in particular for a load- ing cycle with R=0, the maximum effective stress intensity factor may be used for atigue crack growth rate predictions Many theoretical studies are addressed to the problem of modeling the toughening of zirconia-based ceramics through analytical methods. These stud- aim at determining the R-curve of the transforming materials, i.e., the in- crease of the applied stress intensity factor able to promote a stable crack propagation under static loading. Some of these approaches are based on the SO-called weight functions as in Fett (1998)and in Rauchs et al.(1999); and some rely on the theory of dislocations and of the complex potential as shown in the monograph by Karihaloo and Andreasen(1996) or in Moller and Kar chaloo(1995) The phenomenon of the phase transformation in zirconia-based ceramics has been modeled through the formulation of a constitutive law based on the concepts of the mechanics of continua in Sun et al. (1991), Zhang and Lam (1994)and in Lam et al. (1995). The approach proposed by Sun et al. (1991)is adopted to study the strain localization induced by the transformation phenomena in the zirconia ceramics by Yan et al.(1997)

the path of the crack. If this strain field is permanent, i.e., in the case in which the transformation phenomenon is irreversible, a beneficial effect on the fracture toughness is obtained through a reduction of the stress intensity factor. This means that, for a given applied load which would produce an applied stress intensity factor in a non-transforming material, it can be defined an effective stress intensity factor that accounts for the effects given by the transformation strains in transforming materials. The transformation strains in the wake of the crack modifies the stress state around the crack tip in a such a way that the effective stress intensity factor is smaller than the applied stress intensity factor. Moreover, it has been widely proved through experimental tests that the frontal toughening mechanism based on the tetragonal to monoclinic transformation occurring in the Ziconia ceramics gives the material a peculiar cyclic fatigue behaviour especially for small cracks (Steffen et al., 1991). The phenomenon may be described as a two stages process: at the beginning of the fatigue crack propagation process, the crack growth rate exhibits a negative dependency on the applied stress intensity factor; thereafter, after having reached a minimum, a linear positive dependency has been found. In the first stage, the relationship between the applied stress intensity factor and the crack growth rate is strongly dependent on the strength of the transformation capability of the material; whereas, in the second stage the relationship between the applied stress intensity factor and the crack growth rate is linear with a slope which is non-dependent on the transformation strength. This latter is the typical behaviour of long cracks that follows the Paris law like the fatigue crack growth in metals. From the results of the numerical experiments and the comparison with experimental finding presented in Karihaloo and Andreasen (1996), it would appear that the power-law is valid for both small as well as long cracks; in particular for a loading cycle with R = 0, the maximum effective stress intensity factor may be used for fatigue crack growth rate predictions. Many theoretical studies are addressed to the problem of modeling the toughening of zirconia-based ceramics through analytical methods. These studies aim at determining the R-curve of the transforming materials, i.e., the increase of the applied stress intensity factor able to promote a stable crack propagation under static loading. Some of these approaches are based on the so-called weight functions as in Fett (1998) and in Rauchs et al. (1999); and some rely on the theory of dislocations and of the complex potential as shown in the monograph by Karihaloo and Andreasen (1996) or in Møller and Karihaloo (1995). The phenomenon of the phase transformation in zirconia-based ceramics has been modeled through the formulation of a constitutive law based on the concepts of the mechanics of continua in Sun et al. (1991), Zhang and Lam (1994) and in Lam et al. (1995). The approach proposed by Sun et al. (1991) is adopted to study the strain localization induced by the transformation phenomena in the zirconia ceramics by Yan et al. (1997). P. Vena et al. / International Journal of Plasticity 22 (2006) 895–920 897

P. Vena et al. International Journal of Plasticity 22(2006)895-920 The approach based on the formulation of a constitutive law, in contrast to the weight function approach, provides a general framework which is suitable for implementation into numerical models, like the finite element method, aimed at simulating the mechanical behaviour of transforming materials. Similar ap- proaches are also used for the constitutive modeling of other materials subjected to martensitic phase transformation such as the shape memory alloys. An exam ple of such constitutive formulations starting from a micromechanical approach is the one presented by Cherkaoui et al.(1998). A special plasticity-based constitu tive model for shape memory alloy used within a finite element framework is pre sented by Boyd and Lagoudas(1996a, b) and more recently by ladicola and Sha (2004). As a further example, the work by Auricchio(2001) proposes a integration algorithm for shape memory alloy with potential application in the bioengineering field. Finite element studies of the toughening capability of the zirconia-based ceramics can be found for example in Stam et al.(1994)and in Stam and van der Giessen(1995) The fatigue behaviour of zirconia-based ceramics has been considered in the book by Karihaloo and Andreasen(1996)in which, the effect of the phase transformation on the fatigue propagation of a crack in a semi-infinite space has been investigated An analytical approach based on the dislocation formalism has been used in this context. To the Author's knowledge, no other papers deal with finite element models for fatigue crack growth in zirconia-based ceramics The aim of the present study is to develop a finite element model able to predict the fatigue crack growth rate of zirconia-based ceramics subjected to phase transformations The constitutive framework proposed in Sun et al.(1991)has been suitably implemented into a commercial finite element code within the standard displace- ment-based finite element formulation. The approach presented in Sun et al (1991)leads to a constitutive law that is based on the same formalism as that used for the classical plasticity for metals. This allowed for an easy implementa- tion of the iterative procedure aimed at computing the state of stress and the vol- ume fraction of the transformed material. in an incremental form. and the relevant consistent tangent operator to be used into the iterative solution of the nonlinear equilibrium equations In the present paper, the finite element model has been used for the computational evaluation of the fatigue behaviour of a ceramic beam subjected to a cyclic four point bending loading. The effect of the material transformation on the crack growth rate has been investigated. The paper is organized as follows: in the following section a detailed description of the constitutive model for the zirconia-based ceramics and the relevant finite ele- ment implementation are presented; Section 3 presents the computational procedure that has been implemented for the computation of the applied stress intensity factor and the effective stress intensity factor for the four point bending problem; in the Sec- tion 4 two sets of results are presented: (i) the toughening in terms of effective stress intensity factor and applied stress intensity factor and (ii) the crack growth rate for different values of transformation strength; Section 5 will conclude the paper

The approach based on the formulation of a constitutive law, in contrast to the weight function approach, provides a general framework which is suitable for implementation into numerical models, like the finite element method, aimed at simulating the mechanical behaviour of transforming materials. Similar approaches are also used for the constitutive modeling of other materials subjected to martensitic phase transformation such as the shape memory alloys. An example of such constitutive formulations starting from a micromechanical approach is the one presented by Cherkaoui et al. (1998). A special plasticity-based constitutive model for shape memory alloy used within a finite element framework is presented by Boyd and Lagoudas (1996a,b) and more recently by Iadicola and Shaw (2004). As a further example, the work by Auricchio (2001) proposes a integration algorithm for shape memory alloy with potential application in the bioengineering field. Finite element studies of the toughening capability of the zirconia-based ceramics can be found for example in Stam et al. (1994) and in Stam and van der Giessen (1995). The fatigue behaviour of zirconia-based ceramics has been considered in the book by Karihaloo and Andreasen (1996) in which, the effect of the phase transformation on the fatigue propagation of a crack in a semi-infinite space has been investigated. An analytical approach based on the dislocation formalism has been used in this context. To the Authors knowledge, no other papers deal with finite element models for fatigue crack growth in zirconia-based ceramics. The aim of the present study is to develop a finite element model able to predict the fatigue crack growth rate of zirconia-based ceramics subjected to phase transformations. The constitutive framework proposed in Sun et al. (1991) has been suitably implemented into a commercial finite element code within the standard displacement-based finite element formulation. The approach presented in Sun et al. (1991) leads to a constitutive law that is based on the same formalism as that used for the classical plasticity for metals. This allowed for an easy implementation of the iterative procedure aimed at computing the state of stress and the volume fraction of the transformed material, in an incremental form, and the relevant consistent tangent operator to be used into the iterative solution of the nonlinear equilibrium equations. In the present paper, the finite element model has been used for the computational evaluation of the fatigue behaviour of a ceramic beam subjected to a cyclic four point bending loading. The effect of the material transformation on the crack growth rate has been investigated. The paper is organized as follows: in the following section a detailed description of the constitutive model for the zirconia-based ceramics and the relevant finite element implementation are presented; Section 3 presents the computational procedure that has been implemented for the computation of the applied stress intensity factor and the effective stress intensity factor for the four point bending problem; in the Section 4 two sets of results are presented: (i) the toughening in terms of effective stress intensity factor and applied stress intensity factor and (ii) the crack growth rate for different values of transformation strength; Section 5 will conclude the paper. 898 P. Vena et al. / International Journal of Plasticity 22 (2006) 895–920

P. Vena et al. International Journal of Plasticity 22(2006)895-920 2. Material model 2. Constitutive model In contrast to most conventional ceramics that exhibit linear elastic behaviour up to a stress threshold beyond which the brittle failure occurs, the zirconia-based ceramics exhibit a nonlinear stress-strain relationship. This nonlinear mechanical response is believed to be owed to the phase transformation phenomena that tak place into the tetragonal particles or tetragonal grains within the material. From a phenomenological point of view, this phase transformation can be seen as the onset and development of transformation (inelastic)strains that redistribute the stress into the material, in the same manner as plastic strains do in metals For this reason, the phenomenon is often designated as transformation plasticity even if, from the physical standpoint, transformation strains are different in nat ure from plastic strains. In zirconia-based ceramics, the magnitude of the transformation strains is strictly conia ceramics the tetragonal to monoclinic phase implies a volume increase and a shear deformation, although the deviatoric part of the transformation strains may be significantly less important than the volumetric strain because of the twinning ef- fect that compensates most of the shear deformation(Karihaloo and Andreasen 1996 This section reports a brief expounding of the constitutive model for transforming ceramics that, even if it is not the original part of the paper, is required for sake of clarity of the computational approach presented Section 2.2. It is well known that the formulation of a constitutive law relies on the definition of the length scale of the phenomena for which the material model is conceived. The f-m phase transformation occurring in the zirconia-based ceramics( for example the tetragonal zirconia polycrystal- TZP) involves tetragonal grains having size smaller than I um. It is here assumed that the size of the representative volume ele- ment is large enough, with respect to the length scale of the phase transformation phenomenon, to define meaningful stress and strain averages The constitutive framework introduced by Sun et al.(1991)for materials sub- jected to t-+ m phase transformations, in which terminology and ideas borrowed from classical plasticity theories are conveniently used, is here adopted. In this for mulation, the continuum element contains a large number of transforming particles (later denoted with subscript n) treated as inclusions embedded into an elastic matrix of untransformed material subjected to affine deformations. The volume occupied by the transformed particles is V, whereas the volume occupied by the surrounding elas- tic matrix is y The average stress 2 in the continuum is then defined through the average oper ator(.) E=lo)r=v/odv=f(o)v+(1-n)()r

2. Material model 2.1. Constitutive model In contrast to most conventional ceramics that exhibit linear elastic behaviour up to a stress threshold beyond which the brittle failure occurs, the zirconia-based ceramics exhibit a nonlinear stress–strain relationship. This nonlinear mechanical response is believed to be owed to the phase transformation phenomena that take place into the tetragonal particles or tetragonal grains within the material. From a phenomenological point of view, this phase transformation can be seen as the onset and development of transformation (inelastic) strains that redistribute the stress into the material, in the same manner as plastic strains do in metals. For this reason, the phenomenon is often designated as transformation plasticity even if, from the physical standpoint, transformation strains are different in nature from plastic strains. In zirconia-based ceramics, the magnitude of the transformation strains is strictly related to the lattice parameters of the parent and product phases. In general, for zirconia ceramics the tetragonal to monoclinic phase implies a volume increase and a shear deformation, although the deviatoric part of the transformation strains may be significantly less important than the volumetric strain because of the twinning effect that compensates most of the shear deformation (Karihaloo and Andreasen, 1996). This section reports a brief expounding of the constitutive model for transforming ceramics that, even if it is not the original part of the paper, is required for sake of clarity of the computational approach presented Section 2.2. It is well known that the formulation of a constitutive law relies on the definition of the length scale of the phenomena for which the material model is conceived. The t ! m phase transformation occurring in the zirconia-based ceramics (for example the tetragonal zirconia polycrystal – TZP) involves tetragonal grains having size smaller than 1 lm. It is here assumed that the size of the representative volume element is large enough, with respect to the length scale of the phase transformation phenomenon, to define meaningful stress and strain averages. The constitutive framework introduced by Sun et al. (1991) for materials subjected to t ! m phase transformations, in which terminology and ideas borrowed from classical plasticity theories are conveniently used, is here adopted. In this formulation, the continuum element contains a large number of transforming particles (later denoted with subscript I) treated as inclusions embedded into an elastic matrix of untransformed material subjected to affine deformations. The volume occupied by the transformed particles is VI whereas the volume occupied by the surrounding elastic matrix is Vm. The average stress R in the continuum is then defined through the average operator Æ Æ æ as R ¼ hriV ¼ 1 V Z V r dV ¼ f hriV I þ ð1 f ÞhriV m ; ð1Þ P. Vena et al. / International Journal of Plasticity 22 (2006) 895–920 899

P. Vena et al. International Journal of Plasticity 22(2006)895-920 wheref is the volumetric fraction of transformed material which must be smaller than the total amount of transformable particles volume f. For TZP material, all the grains may be subjected to phase transformation, therefore f= l Microscopic and macroscopic strains can be introduced as e and E, respectively the decomposition into elastic and transformation components is introduced in the following ∈=∈+e Applying the linear elastic constitutive law and the average definition for the strains, under the hypothesis that transformed and untransformed particles have the same microscopic compliance tensor M, one has E=M: E=M: o)y=M: a)r=E)v, whereas, the average definition for the transformation strains yields E=(e)y=f(∈")v which can be decomposed into the dilatation(volumetric) and shear(deviatoric components as follows In the above, epd is the constant lattice volume dilatation during the t-m transformation where epd is a material constant and eps is the shear transformation strain In the paper by Sun(Sun et al, 1991), it is assumed that the shear components of the microscopic transformation strains are proportional to the deviatoric part of the stress tensor in the matrix according to the following relationshi =(c")n=4a in which sm is the stress deviator into the matrix and the scalar om is defined as c=V2岢引 This assumption implies the uncoupling of deviatoric transformation strains and the hydrostatic stress components and it is supported by the strong deviatoric stress- based feature of the shear components of the transformation strains(Sun et al 1991, and references therein) In(7)A is a material constant or function which gives the strength of the con straint given by the surrounding matrix For a the surrounding matrix is subjected to a purely hydrostatic stress, no shear transfor- mation strains occur. Using the Eshelby tensor for a spherical inclusion in an infinite elastic body and the Mori and Tanaka averaging technique one can find the

where f is the volumetric fraction of transformed material which must be smaller than the total amount of transformable particles volume f m. For TZP material, all the grains may be subjected to phase transformation, therefore f m = 1. Microscopic and macroscopic strains can be introduced as and E, respectively; the decomposition into elastic and transformation components is introduced in the following: ¼ e þ tr; E ¼ Ee þ Etr. ð2Þ Applying the linear elastic constitutive law and the average definition for the strains, under the hypothesis that transformed and untransformed particles have the same microscopic compliance tensor M, one has Ee ¼ M : R ¼ M : hriV ¼ hM : riV ¼ he iV ; ð3Þ whereas, the average definition for the transformation strains yields Etr ¼ htriV ¼ f htriV I ; ð4Þ which can be decomposed into the dilatation (volumetric) and shear (deviatoric) components as follows: Etr ¼ Epd þ Eps ¼ f hpdiV I þ f hpsiV I . ð5Þ In the above, pd is the constant lattice volume dilatation during the t ! m transformation h pd ij iV I ¼ pddij; ð6Þ where pd is a material constant and ps is the shear transformation strain. In the paper by Sun (Sun et al., 1991), it is assumed that the shear components of the microscopic transformation strains are proportional to the deviatoric part of the stress tensor in the matrix according to the following relationship: ps ij ¼ hpsidV I ¼ A sM ij rM e ð7Þ in which sM ij is the stress deviator into the matrix and the scalar rM e is defined as rM e ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 sM ij sM ij r . ð8Þ This assumption implies the uncoupling of deviatoric transformation strains and the hydrostatic stress components and it is supported by the strong deviatoric stressbased feature of the shear components of the transformation strains (Sun et al., 1991, and references therein). In (7) A is a material constant or function which gives the strength of the constraint given by the surrounding matrix. For rM e ¼ 0, it must be A = 0 because, when the surrounding matrix is subjected to a purely hydrostatic stress, no shear transformation strains occur. Using the Eshelby tensor for a spherical inclusion in an infinite elastic body and the Mori and Tanaka averaging technique one can find the 900 P. Vena et al. / International Journal of Plasticity 22 (2006) 895–920

relationship between the macroscopic stress and the stress acting on the matrix; the deviatoric and volumetric stress components are: s M ij ¼ Sij fB1h ps ij iV I ; ð9Þ rM m ¼ Rm fB2 pd; ð10Þ in which B1 and B2 are defined through the Eshelby tensor as B1 ¼ 2Gð5m 7Þ 15ð1 mÞ ; B2 ¼ Kð4m 2Þ 1 m . ð11Þ In the above expressions, m, G and K are the Poisson ratio, the tangential stiffness and the bulk modulus of the matrix. A stress-based criterion that determines the condition for the phase transformation to occur in the tetragonal particles is also introduced. It depends on the stress state acting in the matrix according to the following condition: F ðrM e ; rM m Þ ¼ 2 3 ArM e þ 3 pdrM m C0ðT ; f Þ ¼ 0 ð12Þ and accounting for the relationships (9) and (10), a transformation criterion depending on the macroscopic stress can be determined F ðR; f Þ ¼ 2 3 AJðSij fB1h ps ij iV I Þ þ 3 pdðRm fB2 pdÞ C0ðT ; f Þ ¼ 0. ð13Þ In the above expression, the operator J has been introduced JðrijÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 rijrij r . ð14Þ The function C0(T,f) depends on the volume fraction of transformed material f and on the absolute temperature T; it accounts for the driving force required to overcome the resistance due to the friction at the interfaces between crystals, the surface energy and the chemical energy associated with the transformation. It can be generally written as C0 ¼ C þ aB0ð pdÞ 2 f ð15Þ in which C represents an activation energy threshold for the process; whereas, the last term is introduced in order to comply with the experimental observations that have shown an increasing resistance to transformation with increasing volumefraction of transformed material (f). In (15), a is a material constant and B0 is defined as follows: B0 ¼ 4Gð1 þ mÞ 1 m þ Gh2 0ð28 20mÞ 5ð1 mÞ ð16Þ with h0 ¼ A 3pd . ð17Þ P. Vena et al. / International Journal of Plasticity 22 (2006) 895–920 901

P. Vena et al. International Journal of Plasticity 22(2006)895-920 Similarly to the metal plasticity theory, the criterion for phase transformation can be described by means of a geometric representation, named transformation surface, in the principal stress three-dimensional reference system. In this kind of representa- tion, the function F(.) for f=0 is a cone having the axes of revolution lying on the hydrostatic axes. The vertex of the cone lies on the tension side of the hydrostatic axes. For a purely hydrostatic state of stress, the phase transformation occurs when the mean macroscopic stress Em reaches the critical value Eu Co(T,) (18) In the case in which the shear transformation strains e are neglected, i. e, if A=0 i assumed, the transformation surface collapse in a plane(deviatoric plane)having the following analytical expression F(∑,f)=3c叫(m-fB2e叫)-Ca(T,f)=0 and the phase transformation is activated by a tensile hydrostatic stress only. Differently from the theory of metal plasticity, the hardening rule for transform- ing materials do not need to be assumed a priori on the basis of the phenomenology Indeed, the evolution of the transformation surface, as soon as the phase transforma tion occurs and the value of f increases, follows a kinematic hardening rule naturally determined by the back stress ob=fB, (eps)v,+fB2epd 8y and its evolution, defined by the transformation strains increment In the special case of A=0, the transformation surface which is a deviatoric plane translates in the tensile side of the hydrostatic axes, and the value of the critical mean stress Em increases with increasing of the volume fraction of transformed material f ng to Co(T,f)+3fepd B2 The incremental relationship for the shear transformation strain can be written according to(7) as follows: E=f()n2+f()n;=f() and the increment of the total macroscopic strains is Ej= Eut e Similarly to the metal plasticity theory, the time increment of the internal variable f, f can be obtained by enforcing a consistency condition a(ep)y 0

Similarly to the metal plasticity theory, the criterion for phase transformation can be described by means of a geometric representation, named transformation surface, in the principal stress three-dimensional reference system. In this kind of representation, the function F(R,f) for f = 0 is a cone having the axes of revolution lying on the hydrostatic axes. The vertex of the cone lies on the tension side of the hydrostatic axes. For a purely hydrostatic state of stress, the phase transformation occurs when the mean macroscopic stress Rm reaches the critical value Rc m Rc m ¼ C0ðT ; f Þ 3pd . ð18Þ In the case in which the shear transformation strains ps are neglected, i.e., if A = 0 is assumed, the transformation surface collapse in a plane (deviatoric plane) having the following analytical expression: F ðR; f Þ ¼ 3 pdðRm fB2 pdÞ C0ðT ; f Þ ¼ 0 ð19Þ and the phase transformation is activated by a tensile hydrostatic stress only. Differently from the theory of metal plasticity, the hardening rule for transforming materials do not need to be assumed a priori on the basis of the phenomenology. Indeed, the evolution of the transformation surface, as soon as the phase transformation occurs and the value of f increases, follows a kinematic hardening rule naturally determined by the back stress rb ij ¼ fB1hpsiV I þ fB2pddij and its evolution, defined by the transformation strains increments. In the special case of A = 0, the transformation surface which is a deviatoric plane translates in the tensile side of the hydrostatic axes, and the value of the critical mean stress Rc m increases with increasing of the volume fraction of transformed material f according to Rc m ¼ C0ðT ; f Þ þ 3f pd2 B2 3pd . ð20Þ The incremental relationship for the shear transformation strain can be written according to (7) as follows: E_ ps ij ¼ _ f h ps ij iV I þ f _ h ps ij iV I ¼ _ f h ps ij idV I ¼ A _ f sM ij rM e ð21Þ and the increment of the total macroscopic strains is E_ ij ¼ E_ e ij þ E_ tr ij ¼ MijklR_ kl þ _ f pddij þ A sM ij rM e . ð22Þ Similarly to the metal plasticity theory, the time increment of the internal variable f, _ f can be obtained by enforcing a consistency condition F_ ¼ oF oRij R_ ij þ oF of _ f þ oF oh ps ij iV I _ h ps ij iV I ¼ 0 ð23Þ 902 P. Vena et al. / International Journal of Plasticity 22 (2006) 895–920

《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_Finite element analysis of the fatigue

《复合材料 Composites》课程教学资源（学习资料）第五章陶瓷基复合材料_Finite element analysis of the fatigue