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
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: Sy=Sy-fB1(e))v (10) in which Bi and B2 are defined through the Eshelby tensor as BI )2;B,k(4y 2G(5v-7) (11) In the above expressions, v, G and K are the Poisson ratio, the tangential stifness and the bulk modulus of the matrix A stress-based criterion that determines the condition for the phase transforma- tion 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(,m)=3A+3m-C0(7,0=0 and accounting for the relationships(9)and(10), a transformation criterion depend g on the macroscopic stress can be determined F(E,)=54J(S, -B, (eP), )+3eP(Em-SB2 epd)-Co(T, f)=0.(13) In the above expression, the operator has been introduced The function Co(T. 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 writ en as n 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 volume- fraction of transformed material () In(15), a is a material constant and Bo is defined as follows 4G(1+v),Gh6(28-20v) Bo 5(1-v) (16) with (17)
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
P. Vena et al. International Journal of Plasticity 22(2006)895-920 and solving for f one has (43s+3-2 ′=B4F+(2+26)02=B4F+(32+21 The volumetric fraction of the transformed particles f increases until the total mount of transformable particles () has been transformed, therefore it f≥0,f<f, f=0, f=f When f=fmax the material behaves as a linear elastic material with irreversible trains. No reverse transformation has been introduced for the tzP 2. 2. Finite element implementation The implementation of the previously expounded constitutive law into a finite ele ment code requires the definition of the tangent stiffness matrix that relates the incre- ment of macroscopic strain with the increment of the macroscopic stress tensor Similarly to the finite element approach to classical metal plasticity(among the other see Zienkiewicz and Taylor, 1991), an elastic-plastic stifness matrix is defined for the increments in which f>0; the elastic matrix is used iff=0 In this formulation, there is no physical meaning attached to the time variable, therefore, the following approximation has been assumed for the time increment The consistent tangent operator is formulated through the definition of two residual equations. R1=0, R1=M△+△f(e叫1+A5)-△E=0 r2=ΦAS-g△f=0, aE,8=5B,A2+(3B2+aBo)(end) The residual Eqs.(30)and (31) are the discretized version of the Eqs.(22)and(24), respectively The system of residual equations is solved by applying the Newton-Raphson ethod based on the following incremental scheme
and solving for _ f one has: _ f ¼ oF oRij R_ ij 2 3 B1A2 þ ð Þ 3B2 þ aB0 pd2 ¼ A sM ij rM e _ Sij þ 3pdR_ m 2 3 B1A2 þ ð Þ 3B2 þ aB0 pd2 . ð24Þ The volumetric fraction of the transformed particles f increases until the total amount of transformable particles (f m) has been transformed, therefore it is: _ f P 0; f < f m; ð25Þ _ f ¼ 0; f ¼ f m. ð26Þ When f = fmax the material behaves as a linear elastic material with irreversible strains. No reverse transformation has been introduced for the TZP. 2.2. Finite element implementation The implementation of the previously expounded constitutive law into a finite element code requires the definition of the tangent stiffness matrix that relates the increment of macroscopic strain with the increment of the macroscopic stress tensor components. Similarly to the finite element approach to classical metal plasticity (among the other see Zienkiewicz and Taylor, 1991), an elastic–plastic stiffness matrix is defined for the increments in which _ f > 0; the elastic matrix is used if _ f ¼ 0. In this formulation, there is no physical meaning attached to the time variable; therefore, the following approximation has been assumed for the time increment: _ D . ð27Þ The consistent tangent operator is formulated through the definition of two residual equations: R1 ¼ 0; ð28Þ r2 ¼ 0 ð29Þ with R1 ¼ MDR þ Df pdI þ A sM rM e DE ¼ 0; ð30Þ r2 ¼ UDR gDf ¼ 0; ð31Þ and U ¼ oF oRij ; g ¼ 2 3 B1A2 þ ð Þð 3B2 þ aB0 pdÞ 2 . ð32Þ The residual Eqs. (30) and (31) are the discretized version of the Eqs. (22) and (24), respectively. The system of residual equations is solved by applying the Newton–Raphson method based on the following incremental scheme: P. Vena et al. / International Journal of Plasticity 22 (2006) 895–920 903