International Journal of Solids and Structures 47(2010)3214-3220 Contents lists available at Science Direct International Journal of Solids and Structures ELSEVIER journalhomepagewww.elsevier.com/locate/ijsolstr Fundamental formulation for transformation toughening Lifeng Ma SEv Lab, Department of Engineering Mechanics, Xi'an Jiaotong University, 710049, china ARTICLE INFO A BSTRACT In this paper, the transformation toughening problem is addressed in the framework of plane strain. The Received 20 April 2010 fundamental solution for a transformed strain nucleus located in an infinite plane is derived first with eceived in revised form 6 July 201( this solution, the transformed inclusion problems are formulated by a Greens function method, and Available online 10 August 2010 the interaction of a crack tip with a single transformation source is found. On the basis of this solution, the fundamental formulations for toughening arising from martensitic and ferroelastic transformation re formulated also using the Greens function method. Finally, some examples are provided to demon- ransformation toughening undamental solution strate the validity and relevance of the fundamental formulations proposed in the paper. Green's function e 2010 Elsevier Ltd. All rights reserved. 1 Introduction and Evans, 1982; Yang and Zhu, 1998: Yi and gao, 2000: Yi et al. 2001: Li and Yang, 2002; Fischer and Boehm, 2005). Another is There is much experimental evidence that the toughness of the finite element method(FEM).(e.g, Zeng et al, 1999, 2004 some ceramics can be substantially enhanced through the con- Vena et al, 2006). The third approach is a Green s function method trolled use of martensitic transformation(see e.g., Garvie et al., (e.g. Budiansky et al, 1983: Lambropoulos, 1986: Rose, 1987 1975: Claussen, 1976: Gupta et al., 1977: Hannink, 1978: Evans Tsukamoto and Kotousov, 2006). The Eshelby-type approach can and Heuer, 1980: Lange, 1982: Munz and Fett, 1998: Hannink be used for analysis of interactions between a crack tip and a dis- et al, 2000: Rauchs et al, 2001, 2002: Kelly and Rose, 2002: Mag- crete transformed zone, but it is not convenient when multiple nani and Brillante, 2005)or ferroelastic transformation(Clussen transformed zones are involved. FEM can be used effectively for et al, 1984). In the case of martensitic transformation, experiments arbitrary complex geometry of transformed zones and also with reveal that the toughening is due to a crack tip stress induced complex material constitutive laws, but multiple analyses must phase transformation. When the stresses in the region near the be undertaken to obtain adequate coverage of the appropriate crack tip reach a critical value, zirconia inclusions particle trans parameter range. The Greens function method is convenient and form from tetragonal to monoclinic, accompanied by a volume straightforward when used for many kinds of geometries of trans- increment of 4% and a shear strain of 16%. These strains induce fur- formed regions. In this paper, we concentrate on the Greens func ther stress in the crack tip region, and the stress intensity factors at tion method to provide improved tools for the analysis of the crack tip may be reduced. Therefore, the fracture toughness of transformation toughening. the ceramic is effectively enhanced, since it takes higher applied Considerable progress has been achieved in the application of loads to raise the stress intensity factor back up to the critical level Greens function methods to transformation toughening probler required to cause continued crack propagation. In contrast to the For example, Hutchinson(1974) solved the plane problem of the martensitic case, ferroelastic transformations typically have only interaction of a semi-infinite crack in an infinite body and two a shear component. Ferroelastic toughening is attributed to do- transformed circular"spots"symmetrically located relative to the nain switching in the crack front and crack wake, inducing stress crack plane. Based on this solution, Budiansky et al.(1983)ob- intensity factor reductions(see, e.g. Yang and Zhu, 1998: Wang tained results for the problem of a contin et al, 2004; Jones et al., 2005. 2007; Jones and Hoffman, 2006: surrounding the crack tip with a focus on the effect of its dilatation. Pojprapai et al., 2008) Thereafter, this approach has been used frequently to study trans In addition to experimental assessments, transformation tough- formation toughening(e. g, Lambropoulos, 1986: Tsukamoto and ening has been also the subject of numerous modeling studies. Kotousov, 2006). However, these solutions are only good for the Three main approaches to model transformation toughening have problems with transformed zone that are symmetrical with respect been used. One is an Eshelby-type approach (e. g. McMeeking to the crack plane. Separately, Rose( 1987)represented both dila ant and deviatoric transformed strain components with a set of Tel.:+862982663861;fax:+862982668751 fundamental singular solutions such as a force-doublet, similar to E-mailaddress:malf@.xjtu.edu.cn the work of Love(1927). His methodology is rigorous, but not 0020-7683/s- see front matter o 2010 Elsevier Ltd. All rights reserved o:10.1016 ijsolstr:201008002
Fundamental formulation for transformation toughening Lifeng Ma * S&V Lab, Department of Engineering Mechanics, Xi’an Jiaotong University, 710049, China article info Article history: Received 20 April 2010 Received in revised form 6 July 2010 Available online 10 August 2010 Keywords: Transformation toughening Fundamental solution Green’s function abstract In this paper, the transformation toughening problem is addressed in the framework of plane strain. The fundamental solution for a transformed strain nucleus located in an infinite plane is derived first. With this solution, the transformed inclusion problems are formulated by a Green’s function method, and the interaction of a crack tip with a single transformation source is found. On the basis of this solution, the fundamental formulations for toughening arising from martensitic and ferroelastic transformation are formulated also using the Green’s function method. Finally, some examples are provided to demonstrate the validity and relevance of the fundamental formulations proposed in the paper. 2010 Elsevier Ltd. All rights reserved. 1. Introduction There is much experimental evidence that the toughness of some ceramics can be substantially enhanced through the controlled use of martensitic transformation (see e.g., Garvie et al., 1975; Claussen, 1976; Gupta et al., 1977; Hannink, 1978; Evans and Heuer, 1980; Lange, 1982; Munz and Fett, 1998; Hannink et al., 2000; Rauchs et al., 2001, 2002; Kelly and Rose, 2002; Magnani and Brillante, 2005) or ferroelastic transformation (Clussen et al., 1984). In the case of martensitic transformation, experiments reveal that the toughening is due to a crack tip stress induced phase transformation. When the stresses in the region near the crack tip reach a critical value, zirconia inclusions particle transform from tetragonal to monoclinic, accompanied by a volume increment of 4% and a shear strain of 16%. These strains induce further stress in the crack tip region, and the stress intensity factors at the crack tip may be reduced. Therefore, the fracture toughness of the ceramic is effectively enhanced, since it takes higher applied loads to raise the stress intensity factor back up to the critical level required to cause continued crack propagation. In contrast to the martensitic case, ferroelastic transformations typically have only a shear component. Ferroelastic toughening is attributed to domain switching in the crack front and crack wake, inducing stress intensity factor reductions (see, e.g. Yang and Zhu, 1998; Wang et al., 2004; Jones et al., 2005, 2007; Jones and Hoffman, 2006; Pojprapai et al., 2008). In addition to experimental assessments, transformation toughening has been also the subject of numerous modeling studies. Three main approaches to model transformation toughening have been used. One is an Eshelby-type approach (e. g. McMeeking and Evans, 1982; Yang and Zhu, 1998; Yi and Gao, 2000; Yi et al., 2001; Li and Yang, 2002; Fischer and Boehm, 2005). Another is the finite element method (FEM), (e.g., Zeng et al., 1999, 2004; Vena et al., 2006). The third approach is a Green’s function method (e.g. Budiansky et al., 1983; Lambropoulos, 1986; Rose, 1987; Tsukamoto and Kotousov, 2006). The Eshelby-type approach can be used for analysis of interactions between a crack tip and a discrete transformed zone, but it is not convenient when multiple transformed zones are involved. FEM can be used effectively for arbitrary complex geometry of transformed zones and also with complex material constitutive laws, but multiple analyses must be undertaken to obtain adequate coverage of the appropriate parameter range. The Green’s function method is convenient and straightforward when used for many kinds of geometries of transformed regions. In this paper, we concentrate on the Green’s function method to provide improved tools for the analysis of transformation toughening. Considerable progress has been achieved in the application of Green’s function methods to transformation toughening problems. For example, Hutchinson (1974) solved the plane problem of the interaction of a semi-infinite crack in an infinite body and two transformed circular ‘‘spots” symmetrically located relative to the crack plane. Based on this solution, Budiansky et al. (1983) obtained results for the problem of a continuum transformation zone surrounding the crack tip with a focus on the effect of its dilatation. Thereafter, this approach has been used frequently to study transformation toughening (e.g., Lambropoulos, 1986; Tsukamoto and Kotousov, 2006). However, these solutions are only good for the problems with transformed zone that are symmetrical with respect to the crack plane. Separately, Rose (1987) represented both dilatant and deviatoric transformed strain components with a set of fundamental singular solutions such as a force-doublet, similar to the work of Love (1927). His methodology is rigorous, but not 0020-7683/$ - see front matter 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2010.08.002 * Tel.: +86 29 82663861; fax: +86 29 82668751. E-mail address: malf@mail.xjtu.edu.cn International Journal of Solids and Structures 47 (2010) 3214–3220 Contents lists available at ScienceDirect International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
L Ma/Intemational Journal of solids and Structures 47(2010)3214- straightforward and is inconvenient for application. Using Rose's 2.2. Muskhelishvili potentials for a transformed strain nucleus solution, a detailed study on transformation toughing has been conducted recently(Karihaloo and Andreasen, 1996). In the pres Consider a differential element with an area da(- dxodyo), which ent paper we carry out a thorough and systematic study of the undergoes an unconstrained irreversible transformation with two Greens function method for formulating problems of transforma- principal strains Exo and Eyo expressed in local principal coordinates tion toughening to enable more powerful tools for that application. Xo. yo as shown in Fig. 1. The origin of the local coordinate system The aim of the present study is to develop new fund Xo, yo lies at s in the global coordinate system x, y, and y is the solutions, from which modeling of transformation toughening by entation angle of the Xo axis (associated with the principal strain Greens function methods can be easily performed. Dilatant and Exo) with respect to the global x-coordinate axis. Next, given the deviatoric strain components of transformation are to be derived physical meaning of an edge dislocation, an infinitesimal element and analyzed in full. with transformation strain can be represented by an assembly of formulation is constructed in the following steps. Firstly, a four dislocations as shown in Fig. 1. The potentials for the four dis fundamental solution for a transformed strain nucleus located locations in the global coordinate system can be written as: within an infinite plane is derived in Section 2. Based on this solu- tion, the transformed inclusions problems are formulated first in (z)=F B 3,(2)=F251-FB1 Section 3. and the interaction of a transformed strain nucleus with a semi-infinite crack is studied in Section 4. Subsequently in Sec B2 By tion 5, fundamental formulations for transformation toughening 中2(2)=F for martensitic and ferroelastic phenomena are developed with the Green,'s function method. To demonstrate the validity and rel- %, (2)=F-B3-,2,(z)=FBa(S3=$)-F-B evance of these formulations, some simple but typical transforma- tion toughening examples are studied in Section 6, and finally conclusions are drawn in Section 7 中4(2)= B44-S4)EB4 where 2 Fundamental solution for a transformed strain nucleus located in an infinite plane solid B1=e(Ex dxo), B3 =ee(Gxo dxo B2=ete(Ey dyo), Ba=eie%(Ey dyo m11m3mmH5=“(②)s=+(② nucleus will be modeled in terms of the mathematical edge dislo- S2=S S4=S+ The subscripts on dislocation parameters 2.1. Muskhelishvili formulation and Muskhelishvili potentials for an above expressions refer to the denoted disl ntials in the as numbered in Fig. 1. The corresponding Burgers vector In the Muskhelishvili complex formulation of plane elasticity. ecial meaning wi deformation af component s o t stress and disp acements areexpressedin temos The potentials for an infinite plane due to a transformation Muskhelishvili, 1953), strain with principal values Exo, yo, oriented in the direction shown 01+022=2{中(2)+中(z) 02-i012=中(2)+g(2)+(z-2)(2) (2.1) 2(u11+i21)=K更(2)-(2)-(-2)中(2) here, i=V-1, Z=X1+ix2, p (z)=dp(z)/dz, u is shear modulus, 3-4v for plane strain, the comma followed by a subscript i indicates differentiation with respect to x, and the bar over a func- tion denotes its complex conjugate. It is known that the muskh lishvili potentials for an edge dislocation with Burgers vector B with magnitude b, located at point s within an infinite plane solid, d can be expressed as(Suo, 1989): (2)=FB 9(2)=FB (-s) 22) From the results above, we will derive the potentials of a trans formed strain nucleus as follows Fig. 1. A concentrated transformed strain located in an infinite plane solid
straightforward and is inconvenient for application. Using Rose’s solution, a detailed study on transformation toughing has been conducted recently (Karihaloo and Andreasen, 1996). In the present paper we carry out a thorough and systematic study of the Green’s function method for formulating problems of transformation toughening to enable more powerful tools for that application. The aim of the present study is to develop new fundamental solutions, from which modeling of transformation toughening by Green’s function methods can be easily performed. Dilatant and deviatoric strain components of transformation are to be derived and analyzed in full. The formulation is constructed in the following steps. Firstly, a fundamental solution for a transformed strain nucleus located within an infinite plane is derived in Section 2. Based on this solution, the transformed inclusions problems are formulated first in Section 3, and the interaction of a transformed strain nucleus with a semi-infinite crack is studied in Section 4. Subsequently in Section 5, fundamental formulations for transformation toughening for martensitic and ferroelastic phenomena are developed with the Green’s function method. To demonstrate the validity and relevance of these formulations, some simple but typical transformation toughening examples are studied in Section 6, and finally conclusions are drawn in Section 7. 2. Fundamental solution for a transformed strain nucleus located in an infinite plane solid In this section, we seek the Muskhelishvili potentials for a transformed strain nucleus located in an infinite plane solid. The strain nucleus will be modeled in terms of the mathematical edge dislocation solutions. 2.1. Muskhelishvili formulation and Muskhelishvili potentials for an edge dislocation In the Muskhelishvili complex formulation of plane elasticity, all components of stress and displacements are expressed in terms of two potential functions, U(z) and X(z) as follows (Suo, 1989; Muskhelishvili, 1953), r11 þ r22 ¼ 2½UðzÞ þ UðzÞ r22 ir12 ¼ ½UðzÞ þ XðzÞþðz zÞU0 ðzÞ 2lðu1;1 þ iu2;1Þ ¼ jUðzÞ XðzÞðz zÞU0 ðzÞ ð2:1Þ where, i ¼ ffiffiffiffiffiffiffi 1 p , z = x1 + ix2, U0 (z) = dU(z)/dz, l is shear modulus, j = 3 4m for plane strain, the comma followed by a subscript i indicates differentiation with respect to xi, and the bar over a function denotes its complex conjugate. It is known that the Muskhelishvili potentials for an edge dislocation with Burgers vector B with magnitude b, located at point s within an infinite plane solid, can be expressed as (Suo, 1989): UðzÞ ¼ F B z s XðzÞ ¼ FB ðs sÞ ðz sÞ 2 F B z s B ¼ beiw F ¼ l pið1 þ jÞ ð2:2Þ From the results above, we will derive the potentials of a transformed strain nucleus as follows. 2.2. Muskhelishvili potentials for a transformed strain nucleus Consider a differential element with an area dA(= dx0dy0), which undergoes an unconstrained irreversible transformation with two principal strains ex0 and ey0 expressed in local principal coordinates x0, y0 as shown in Fig. 1. The origin of the local coordinate system x0, y0 lies at s in the global coordinate system x, y, and w is the orientation angle of the x0 axis (associated with the principal strain ex0) with respect to the global x-coordinate axis. Next, given the physical meaning of an edge dislocation, an infinitesimal element with transformation strain can be represented by an assembly of four dislocations as shown in Fig. 1. The potentials for the four dislocations in the global coordinate system can be written as: U1ðzÞ ¼ F B1 z s1 ; X1ðzÞ ¼ F B1ðs1 s1Þ ðz s1Þ 2 F B1 z s1 U2ðzÞ ¼ F B2 z s2 ; X2ðzÞ ¼ F B2ðs2 s2Þ ðz s2Þ 2 F B2 z s2 U3ðzÞ ¼ F B3 z s3 ; X1ðzÞ ¼ F B3ðs3 s3Þ ðz s3Þ 2 F B3 z s3 U4ðzÞ ¼ F B4 z s4 ; X4ðzÞ ¼ F B4ðs4 s4Þ ðz s4Þ 2 F B4 z s4 ð2:3Þ where B1 ¼ eiwðex0 dx0Þ; B3 ¼ eipeiwðex0 dx0Þ; B2 ¼ ei p 2 eiwðey0 dy0Þ; B4 ¼ ei p 2 eiwðey0 dy0Þ; s1 ¼ s eiwei p 2 dy0 2 ; s3 ¼ s þ eiwei p 2 dy0 2 s2 ¼ s eiw dx0 2 ; s4 ¼ s þ eiw dx0 2 ð2:4Þ The subscripts on dislocation parameters and potentials in the above expressions refer to the denoted dislocation as numbered in Fig. 1. The corresponding Burgers vector Bi in (2.4) is bestowed with a special meaning which represents the residual deformation of the differential element due to transformation. The potentials for an infinite plane due to a transformation strain with principal values ex0, ey0, oriented in the direction shown Fig. 1. A concentrated transformed strain located in an infinite plane solid. L. Ma / International Journal of Solids and Structures 47 (2010) 3214–3220 3215
L Ma/International Journal of solids and Structures 47(2010)3214-3220 in Fig. 1 within the differential element area da= dxodyo, can be ob-3.1. Influence function for transformed inclusions tained by superposing the four potentials together as 中(2)=1(2)+中3(2)+2(2)+中4(2) ubmitting Eq.(2. 8)into Eg. (2. 1) we may obtain the stress 9(2)=1(2)+93(2)+[92(2)+94(2 (2.5) components denoted as of(z, s),ci2(z, s)and o2(z, s)at point z, due to the presence of a point transformed region at positi After a lengthy but straightforward manipulation, we get We denote these influence functions by p(z)=iFe (z -dody f1(x.yx,y3)=01(2,s) 2(xyx,y)=02(z,s) f3(x,y: Xs, ys )=012(z,s) g(2)=iF2(o+80)-ew(o-2o)2c2w(o-a so-called influence functions in Eqs. ( 3. 1)are used to calculate the ∫20+y0)-e2w(0-ao) Ke(z-s)(-s5dxodyo where z=x+y and s-x,+iy, in the global coordinate system.The stress field due to presence of transformed inclusions in an infinite (26) Clearly the potentials in Eq(2.6)are of infinitesimal order. Finite 3. 2. Formulation of the stress field due to transformed inclusions by the order potentials for the effect per unit area thus characterize the Green's function method transformed strain nucleus, located at s in an infinite plane and are d(2)=iFe2(0-e2o) Now consider an infinite plane solid in which some portions suffer transformation due to external load( Fig. 2)and the trans- formed region corresponds to area A= Ek- Ak. Generally speaking, a)=r20+c00+2-1001-69 the transformed particles could not fully occupy the nominal region A, as in practice the material transformation does not run to full (2.7) 100%. To reflect this, we define a transformation density Junction For convenience, we may rewrite Eq (2.7)in a compact form area, using the value in the range 0< D(xs. ys)<1. Then the expres- sion for the stress at any point(x, y) in the plane can be formulated 中0(2) (28) 2-/2×d on1(x,y)=//f(x,y: xs, ys)D(xs, ys )dx, dys 022(x,y)=///2(x,y: x, ys)D(xs, ys)dx,dys (3.2) C=iFe v G12(X C2=iF{2(c0+40)-e2o(o-no (29) f3(x, y; xs, ys)D(xs, ys)dx dys, Xs, ysE A fluence functions fi(x,y: xs, ys) f2(x,y: xs, ys), 3(x,y:xs, ys) The set of potentials given by(2. 8).(2.)gives the fundamental 3. 1). If the transformation density function D(xs, ys )is solution for a transformation-induced strain nucleus in an infinite stress field will depend only on the influence function plane. This is the basic result on which the further analysis pre sformation zone shape and extent. Eq (3. 2)is the fun sented in this paper is built. lution for the transformed inclusion probler When Eyo=Exo, namely, for a purely dilatational transformation, Eq (2.8)will be independent of the orientation angle consistent with the expected isotropy of the effects of such a transformation. On the other hand, when Gyo--Exo, the solution for purely shear transformation is reached. It can be easily proved that the solutions of the two extreme cases degenerated from (2.8)are consistent with the ones obtained by rose( 1987). 3. Formulation of transformed inclusion problems by the Green's function method In this section we will use the fundamental solution in the above Section to formulate the transformed inclusion problems. Before doing so, we must emphasize that the following is a contin- uum description of polycrystalline systems (e.g. ceramics) and is based on averaging over sufficient numbers of grains within the matrix, including transformed particles. Also, we assume that the elastic properties of the transformed zone(inclusion) are identical to those prior to the transformation, following McMeeking and Evans(1982), and Budiansky et al. (1983). Fig. 2. Transformation strain zones in an infinite plane
in Fig. 1 within the differential element area dA = dx0dy0, can be obtained by superposing the four potentials together as UðzÞ¼½U1ðzÞ þ U3ðzÞ þ ½U2ðzÞ þ U4ðzÞ XðzÞ¼½X1ðzÞ þ X3ðzÞ þ ½X2ðzÞ þ X4ðzÞ ð2:5Þ After a lengthy but straightforward manipulation, we get UðzÞ ¼iFe2iw ðey0 ex0Þ ðzsÞ 2 dx0dy0 ¼iFe2iw ðey0 ex0Þ ðzsÞ 2 dA XðzÞ ¼iF 2ðex0 þey0Þe2iwðey0 ex0Þ ðzsÞ 2 þ 2e2iwðey0 ex0Þ ðzsÞ 3 ðssÞ ( )dx0dy0 ¼iF 2ðex0 þey0Þe2iwðey0 ex0Þ ðzsÞ 2 þ 2e2iwðey0 ex0Þ ðzsÞ 3 ðssÞ ( )dA ð2:6Þ Clearly the potentials in Eq. (2.6) are of infinitesimal order. Finite order potentials for the effect per unit area thus characterize the transformed strain nucleus, located at s in an infinite plane, and are U0ðzÞ ¼ iFe2iw ðey0 ex0Þ ðz sÞ 2 X0ðzÞ ¼ iF 2ðex0 þ ey0Þ e2iwðey0 ex0Þ ðz sÞ 2 þ 2e2iwðey0 ex0Þ ðz sÞ 3 ðs sÞ ( ) ð2:7Þ For convenience, we may rewrite Eq. (2.7) in a compact form as U0ðzÞ ¼ C1 ðz sÞ 2 X0ðzÞ ¼ C2 ðz sÞ 2 þ C3 ðs sÞ ðz sÞ 3 ð2:8Þ where C1 ¼ iFe2iw ðey0 ex0Þ C2 ¼ iF½2ðex0 þ ey0Þ e2iwðey0 ex0Þ C3 ¼ 2iFe2iw ðey0 ex0Þ ð2:9Þ The set of potentials given by (2.8), (2.9) gives the fundamental solution for a transformation-induced strain nucleus in an infinite plane. This is the basic result on which the further analysis presented in this paper is built. When ey0 = ex0, namely, for a purely dilatational transformation, Eq. (2.8) will be independent of the orientation angle w, consistent with the expected isotropy of the effects of such a transformation. On the other hand, when ey0 = ex0, the solution for purely shear transformation is reached. It can be easily proved that the solutions of the two extreme cases degenerated from (2.8) are consistent with the ones obtained by Rose (1987). 3. Formulation of transformed inclusion problems by the Green’s function method In this section, we will use the fundamental solution in the above Section to formulate the transformed inclusion problems. Before doing so, we must emphasize that the following is a continuum description of polycrystalline systems (e.g. ceramics) and is based on averaging over sufficient numbers of grains within the matrix, including transformed particles. Also, we assume that the elastic properties of the transformed zone (inclusion) are identical to those prior to the transformation, following McMeeking and Evans (1982), and Budiansky et al. (1983). 3.1. Influence function for transformed inclusions Submitting Eq. (2.8) into Eq. (2.1) we may obtain the stress components denoted as r0 11ðz; sÞ;r0 12ðz; sÞ and r0 22ðz; sÞ at point z, due to the presence of a point transformed region at position s. We denote these influence functions by f1ðx; y; xs; ysÞ ¼ r0 11ðz; sÞ f2ðx; y; xs; ysÞ ¼ r0 22ðz; sÞ f3ðx; y; xs; ysÞ ¼ r0 12ðz; sÞ ð3:1Þ where z = x + iy and s = xs + iys in the global coordinate system. The so-called influence functions in Eqs. (3.1) are used to calculate the stress field due to presence of transformed inclusions in an infinite plane solid. 3.2. Formulation of the stress field due to transformed inclusions by the Green’s function method Now consider an infinite plane solid in which some portions suffer transformation due to external load (Fig. 2) and the transformed region corresponds to area A ¼ PN k¼1Ak. Generally speaking, the transformed particles could not fully occupy the nominal region A, as in practice the material transformation does not run to full 100%. To reflect this, we define a transformation density function D(xs,ys), to describe the extent of transformation in the nominal area, using the value in the range 0 6 D(xs,ys) 6 1. Then the expression for the stress at any point (x,y) in the plane can be formulated by the Green’s function method as r11ðx; yÞ ¼ Z Z A f1ðx; y; xs; ysÞDðxs; ysÞdxsdys r22ðx; yÞ ¼ Z Z A f2ðx; y; xs; ysÞDðxs; ysÞdxsdys r12ðx; yÞ ¼ Z Z A f3ðx; y; xs; ysÞDðxs; ysÞdxsdys; xs; ys 2 A ð3:2Þ where the influence functions f1(x,y;xs,ys), f2(x,y;xs,ys), f3(x,y; xs,ys) are given by (3.1). If the transformation density function D(xs,ys) is a constant, the stress field will depend only on the influence function and the transformation zone shape and extent. Eq. (3.2) is the fundamental solution for the transformed inclusion problems. Fig. 2. Transformation strain zones in an infinite plane. 3216 L. Ma / International Journal of Solids and Structures 47 (2010) 3214–3220
L Ma/Intemational Journal of Solids and Structures 47(2010)3214-322 4. a transformed strain nucleus interacting with a semi-infinite C2 (3-s) In this section the basic problem of a transformed strain nucleu Q(2)=2√1-C(5-)55+ crack length and other dimensions. The solution will be used to ob- ((1/1V2+C3(3-s)-MMBhva 23 that the size of the transformed area is small compared with the C1 tain the influence functions to formulate the problems about tran formation toughening in Section 5. (4.3) 4.1. Complex po ls for the model 4.2.Influence function for stress intensity factors Consider a transformed strain source located at s ha orientation angle y, interacting with a semi-infinite crack By virtue of the second equation of (2.1). the definition of the in Fig. 3. In the previous section, we obtained the stress intensity factor(SIF)become potentials po(, s). @o(z, s) for a source in an infinite plane solid without a crack. Using the superposition principle and following K=K,+iKu= lim v2Tx(o2+ional the procedure described previously (Ma et al, 2006), one can find the exact complex potentials for the interaction of a trans- imv2x两x)+(x lengthy details, we present the general form of the result as Inserting Eq. (4.3)into (4. 4), we obtain the siF due to the presence follows 9()= =0+01a dt}+920(z K=K1+in=2C1=+C2 中(z)= po(t)+20(0)]Vci (4.1) dt(+ o( K=K+iKn Direct substitution of Eq (2.8)into Eq. (4.1) gives 2(1+k) (0-a0)(5-s) C1 C2 (4.6) 92(z)= (-3)2(t-s)2(t-s Since the stress intensity factors(4.5)or (4.6)are induced by a single point source of transformation(namely, a strain nucleus ) the SIF due to the presence of extended transformed areas can be calcu- lated by integrating the transformation strain contribution over he source region, i.e. the transformation zone area. The expression in Eq (4.6)will play the role of the integral kernel. We denote this 中(z) point influence function for the siF by C f4(s)= 「=w=+1=x After some manipulation, the complex functions for the interaction of a transformation strain source with a semi-infinite crack are fi- 5. Formulation of the transformation tou ng problems nally obtained as: with the green's function method Transformation-induced strain influence functions of different kinds have been derived in the previous sections. In this section, we use the influence functions to formulate the transformation toughening problems with Greens function method. 5.1. Martensitic transformation toughening The problem geometry is shown in Fig. 4. As before, the transformation density function over the transformed zone due to the presence of the transformation zone is given by △K=△K1+△K Here fa(xs, ys)=fa(s) given in Eq(4.7). Generally, when remote loading is applied so as to create stress ty Koo, the material in the vicinity of Fig 3. A transformation strain source interacting with a semi-infinite crack. transformation zone and the transformation sti ck即 s the
4. A transformed strain nucleus interacting with a semi-infinite crack In this section the basic problem of a transformed strain nucleus interacting with a semi-infinite crack will be studied. It is assumed that the size of the transformed area is small compared with the crack length and other dimensions. The solution will be used to obtain the influence functions to formulate the problems about transformation toughening in Section 5. 4.1. Complex potentials for the model Consider a transformed strain source located at s having the orientation angle w, interacting with a semi-infinite crack shown in Fig. 3. In the previous section, we obtained the complex potentials U0(z, s), X0(z, s) for a source in an infinite plane solid without a crack. Using the superposition principle and following the procedure described previously (Ma et al., 2006), one can find the exact complex potentials for the interaction of a transformation strain source with a semi-infinite crack. Omitting lengthy details, we present the general form of the result as follows: XðzÞ ¼ 1 ffiffi z p 1 2p Z 0 1 U0ðtÞ þ X0ðtÞ ffiffiffiffiffi jtj p t z dt ( ) þ X0ðzÞ UðzÞ ¼ 1 ffiffi z p 1 2p Z 0 1 U0ðtÞ þ X0ðtÞ ffiffiffiffiffi jtj p t z dt ( ) þ U0ðzÞ ð4:1Þ Direct substitution of Eq. (2.8) into Eq. (4.1) gives XðzÞ ¼ 1 ffiffi z p 1 2p Z 0 1 C1 ðt sÞ 2 þ C2 ðt sÞ 2 þ C3 ðs sÞ ðt sÞ 3 " # ffiffiffiffiffi jtj p 1 t z dt ( ) þ C2 ðz sÞ 2 þC3 ðs sÞ ðz sÞ 3 UðzÞ ¼ 1 ffiffi z p 1 2p Z 0 1 C1 ðt sÞ 2 þ C2 ðt sÞ 2 C3 ðs sÞ ðt sÞ 3 " # ffiffiffiffiffi jtj p 1 t z dt ( ) þ C1 ðzsÞ 2 ð4:2Þ After some manipulation, the complex functions for the interaction of a transformation strain source with a semi-infinite crack are fi- nally obtained as: XðzÞ ¼ 1 2 ffiffi z p C1 2 ffi s p ffi s p þ ffiffi z p ð Þ2 þ C2 2 ffi s p ffi s p þ ffiffi z p ð Þ2 C3ðs sÞ 3 ffi s p þ ffiffi z p ð Þ 8ðsÞ 3 2 ffi s p þ ffiffi z p ð Þ3 8 >>>: 9 >>= >>; þ C2 ðz sÞ 2 þ C3 ðs sÞ ðz sÞ 3 UðzÞ ¼ 1 2 ffiffi z p C1 2 ffi s p ffi s p þ ffiffi z p ð Þ2 þ C2 2 ffi s p ffi s p þ ffiffi z p ð Þ2 þC3ðs sÞ 3 ffi s p þ ffiffi z p ð Þ 8ðsÞ 3 2 ffi s p þ ffiffi z p ð Þ3 8 >>>: 9 >>= >>; þ C1 ðz sÞ 2 ð4:3Þ 4.2. Influence function for stress intensity factors By virtue of the second equation of (2.1), the definition of the stress intensity factor (SIF) becomes, K ¼ KI þ iKII ¼ lim x!0þ ffiffiffiffiffiffiffiffiffi 2px p ½r22 þ ir12 ¼ lim x!0þ ffiffiffiffiffiffiffiffiffi 2px p ½UðxÞ þ XðxÞ ð4:4Þ Inserting Eq. (4.3) into (4.4), we obtain the SIF due to the presence of the strain source as K ¼ KI þ iKII ¼ ffiffiffiffiffiffi 2p p 2 C1 1 s ffiffi s p þ C2 1 s ffiffi s p C3ðs sÞ 3 ffiffi s p 4s3 ð4:5Þ or K ¼ KI þ iKII ¼ l ffiffiffiffiffiffi 2p p ð1 þ jÞ e2iwðey0ex0Þ s ffi s p þ 2ðex0þey0Þe2iw ½ ðey0ex0Þ s ffi s p 2e2iwðey0 ex0Þðs sÞ 3 ffi s p 4s3 2 4 3 5 ð4:6Þ Since the stress intensity factors (4.5) or (4.6) are induced by a single point source of transformation (namely, a strain nucleus), the SIF due to the presence of extended transformed areas can be calculated by integrating the transformation strain contribution over the source region, i.e. the transformation zone area. The expression in Eq. (4.6) will play the role of the integral kernel. We denote this point influence function for the SIF by f4ðsÞ ¼ l ffiffiffiffiffiffi 2p p ð1 þ jÞ e2iwðey0ex0Þ s ffi s p þ 2ðex0þey0Þe2iwðey0ex0 ½ Þ s ffi s p 2e2iwðey0 ex0Þðs sÞ 3 ffi s p 4s3 2 4 3 5 ð4:7Þ 5. Formulation of the transformation toughening problems with the Green’s function method Transformation-induced strain influence functions of different kinds have been derived in the previous sections. In this section, we use the influence functions to formulate the transformation toughening problems with Green’s function method. 5.1. Martensitic transformation toughening The problem geometry is shown in Fig. 4. As before, the transformation density function over the transformed zone A ¼ PN k¼1Ak is denoted by D(xs,ys). The net SIF at the crack tip due to the presence of the transformation zone is given by DK ¼ DKI þ DKII ¼ Z Z A f4ðxs; ysÞDðxs; ysÞdA ð5:1Þ Here f4(xs,ys) = f4(s) given in Eq. (4.7). Generally, when remote loading is applied so as to create stress intensity K1, the material in the vicinity of the crack tip may undergo transformation under the high local stresses. Assuming the Fig. 3. A transformation strain source interacting with a semi-infinite crack. transformation zone and the transformation strain are known, L. Ma / International Journal of Solids and Structures 47 (2010) 3214–3220 3217
L Ma/ International Journal of solids and Structures 47(2010)3214-3220 A K K Fig 4. Transformed zone in front of a crack tip under remote load g(5.1), allows the evaluation of the crack tip"transformation or hielding"effect in terms of SIF reduction. This allows the stability ∫(r,0,ψ) 2(Bs0-24)-3c0s(0-2) +k(+iin(0-24)+3sin(0-2) 5.2. Ferroelastic transformation toughening (5.4) The toughening mechanism of ferroelastic transformation re- Similarly, if the transformed zone is known as shown in Fig 4, the sults from the 90 ferroelastic domain switching(interchanging net SiF due to domain switching can be computed by the Green's of the ferroelastic long and short axes)(Jones et al, 2007 ). This pro- function methe cedure can be briefly illustrated with Fig. 5. The mechanism of ferroelastic transformation is different from AK= AK+iAk / fs(r, 0, )D(, 0)dA martensitic transformation, but, mathematically, this case is a spe- cial transformation toughening case of the above general solution. where D(r, e)is the transformation density function. Let us consider one elementary domain which is involved in the switching phenomenon in the plane strain regime as shown in 6. Examples The domain has an original orientation l. We suppose that residual stress present prior to domain switching. After The purpose of the present paper is to establish a mathematical switching fromψtoψ+ we can find that framework for the solution of direct transformation toughening (5.2) problems, i.e. the computation of the effects of transformation-in- duced strain on the crack tip SIF. The questions about the region where Eo=(c-a)/a寒 ind the extent of transformation are not addressed here. it is clear Substituting (5.2) into(4.7), the influence function for the net however, that in practice, prior to the evaluation of SIF incorporat siF due to the elementary domain switching is obtained: ng the effects of transformation toughening, the transformed re- gion and transformation strain must be determined first. In the 52x+655+2(1-3 (5.3) literature, several criteria have been proposed for material trans formation, such as the mean stress criterion(see e.g. Budiansky Fig. 5. Schematic demonstration of the toughening mechanism due to 90 ferroelastic domain switching near a crack tip: (a) Pre-switching: (b) post-switching(in dashed
Eq. (5.1), allows the evaluation of the crack tip ‘‘transformation shielding” effect in terms of SIF reduction. This allows the stability of the cracked body to be assessed. 5.2. Ferroelastic transformation toughening The toughening mechanism of ferroelastic transformation results from the 90 ferroelastic domain switching (interchanging of the ferroelastic long and short axes) (Jones et al., 2007). This procedure can be briefly illustrated with Fig. 5. The mechanism of ferroelastic transformation is different from martensitic transformation, but, mathematically, this case is a special transformation toughening case of the above general solution. Let us consider one elementary domain which is involved in the switching phenomenon in the plane strain regime as shown in Fig. 6. The domain has an original orientation w. We suppose that there is no residual stress present prior to domain switching. After domain switching from w to w þ p 2, we can find that ex0 ¼ e0; ey0 ¼ e0 ð5:2Þ where e0 = (c a)/a 1. Substituting (5.2) into (4.7), the influence function for the net SIF due to the elementary domain switching is obtained: f5ðsÞ ¼ 2le0 ffiffiffiffiffiffi 2p p ð1 þ jÞ e2iw s ffiffi s p þ 1 2 1 3 s s e2iw s ffiffi s p ð5:3Þ or f5ðr; h; wÞ ¼ le0 ffiffiffiffiffiffi 2p p ð1 þ jÞr 3 2 3 cos 3 2 h 2w 3 cos 7 2 h 2w þi sin 3 2 h 2w þ 3 sin 7 2 h 2w ( ) ð5:4Þ Similarly, if the transformed zone is known as shown in Fig. 4, the net SIF due to domain switching can be computed by the Green’s function method: DK ¼ DKI þ iDKII ¼ Z Z A f5ðr; h; wÞDðr; hÞdA ð5:5Þ where D(r,h) is the transformation density function. 6. Examples The purpose of the present paper is to establish a mathematical framework for the solution of direct transformation toughening problems, i.e. the computation of the effects of transformation-induced strain on the crack tip SIF. The questions about the region and the extent of transformation are not addressed here. It is clear, however, that in practice, prior to the evaluation of SIF incorporating the effects of transformation toughening, the transformed region and transformation strain must be determined first. In the literature, several criteria have been proposed for material transformation, such as the mean stress criterion (see e.g. Budiansky Fig. 4. Transformed zone in front of a crack tip under remote load. Fig. 5. Schematic demonstration of the toughening mechanism due to 90ferroelastic domain switching near a crack tip: (a) Pre-switching; (b) Post-switching (in dashed circle). 3218 L. Ma / International Journal of Solids and Structures 47 (2010) 3214–3220
L Ma/Intemational Journal of Solids and Structures 47(2010)3214-3220 y Pre-switching Post-switc Fig. 6. A 90 switching elementary domain with an original orientation a near the crack ti et al 1983). the shear stress criterion(Evans and Cannon, 1986), and D(xs, ys)=1. Inserting Eq (6.1)into Eq (4.7)we get the transformation strain energy criterion(see, e.g. Lambropoulos, 1986), the energy switching criterion(Hwang et al, 1995)and 4 (s)=-4Eok1 other criteria for transformation toughening, which are reviewed 2π(1+K in the reference (Hannink et al, 2000). According to different mate- Substituting Eq (6.2) into Eq (5.1), we get AK of the crack tip due to be used to evaluate the transformed zone and assess the degree of transformation toughening. While the problem of transforma- AK=AK +iAK ∫4(x,y)D(x,ydA tion criterion remains a controversial issue. its further discussion lies outside the scope of this pape In this section, our aim is to verify the validity of the obtained fundamental solutions and to demonstrate the efficiency of the for- (1+k)ls√」 mulations obtained above. Three simple but representative trans- formation toughening examples will be investigated for given transformation zones. In order to attack the inverse problem of identifying the most appropriate transformation criterion, different This result is completely consistent with the results obtained by transformation zones can be considered and compared using the IcMeeking and Evans(1982). explicit solutions given below. 6.2. Example 2: a semi-infinite crack enclosed by a transformed wake 6.1. Example 1: interaction between an infinite crack and a transformed spot Consider a semi-infinite crack enclosed by a transformed wake as shown in Fig. 8. Suppose the radius of the transformed circular his example has been studied by McMeeking and Evans area in front of crack tip is R, the infinite transformed area is A, through an Eshelby-type approach. So the existing solutions transformation is a dilatational transformation problem can be regarded as a standard solution to verify the D(r, 0)=1. The influence function is identical (6.2)which y of the obtained fundamental solutions in this paper. can be rewritten as R<r, interacts with an infinite crack, as shown in Fig. 7. Suppose f(r, 0)=-46oA-rtcos30-isin2e a transformed circular region of radius R and center at(r, 0). the transformation strain within the region is dilatational (1+K) ExD= Eyo= Eo (6.1) Substituting Eq (6. 4)into Eq. (5. 1). after some straightforward :∠ Fig. 7. Interaction between an infinite crack and a transformed spot(McMeeking Fig 8. A crack is enclosed by a transformed wake
et al., 1983), the shear stress criterion (Evans and Cannon, 1986), the transformation strain energy criterion (see, e.g. Lambropoulos, 1986), the energy switching criterion (Hwang et al., 1995) and other criteria for transformation toughening, which are reviewed in the reference (Hannink et al., 2000). According to different material transformation mechanisms, the corresponding criterion can be used to evaluate the transformed zone and assess the degree of transformation toughening. While the problem of transformation criterion remains a controversial issue, its further discussion lies outside the scope of this paper. In this section, our aim is to verify the validity of the obtained fundamental solutions and to demonstrate the efficiency of the formulations obtained above. Three simple but representative transformation toughening examples will be investigated for given transformation zones. In order to attack the inverse problem of identifying the most appropriate transformation criterion, different transformation zones can be considered and compared using the explicit solutions given below. 6.1. Example 1: interaction between an infinite crack and a transformed spot This example has been studied by McMeeking and Evans (1982) through an Eshelby-type approach. So the existing solutions of the problem can be regarded as a standard solution to verify the validity of the obtained fundamental solutions in this paper. A transformed circular region of radius R and center at (r,h), R r, interacts with an infinite crack, as shown in Fig. 7. Suppose the transformation strain within the region is dilatational, ex0 ¼ ey0 ¼ e0 ð6:1Þ and D(xs,ys) = 1. Inserting Eq. (6.1) into Eq. (4.7) we get f4ðsÞ ¼ 4e0l ffiffiffiffiffiffi 2p p ð1 þ jÞ 1 s ffiffi s p ð6:2Þ Substituting Eq. (6.2) into Eq. (5.1), we get DK of the crack tip due to presence of the transformed spot as DK ¼ DKI þ iDKII ¼ Z Z A f4ðxs; ysÞDðxs; ysÞdA ¼ 4e0lpR2 ffiffiffiffiffiffi 2p p ð1 þ jÞ 1 s ffiffi s p ¼ 4e0lpR2 ffiffiffiffiffiffi 2p p ð1 þ jÞ r3 2 cos 3 2 h i sin 3 2 h ð6:3Þ This result is completely consistent with the results obtained by McMeeking and Evans (1982). 6.2. Example 2: a semi-infinite crack enclosed by a transformed wake Consider a semi-infinite crack enclosed by a transformed wake as shown in Fig. 8. Suppose the radius of the transformed circular area in front of crack tip is R, the infinite transformed area is A, transformation is a dilatational transformation (ex0 = ey0 = e0), and D(r,h) = 1. The influence function is identical to Eq. (6.2) which can be rewritten as f4ðr; hÞ ¼ 4e0l ffiffiffiffiffiffi 2p p ð1 þ jÞ r3 2 cos 3 2 h i sin 3 2 h ð6:4Þ Substituting Eq. (6.4) into Eq. (5.1), after some straightforward manipulation, we get Fig. 6. A 90 switching elementary domain with an original orientation w near the crack tip. Fig. 7. Interaction between an infinite crack and a transformed spot (McMeeking and Evans, 1982). Fig. 8. A crack is enclosed by a transformed wake. L. Ma / International Journal of Solids and Structures 47 (2010) 3214–3220 3219
L Ma/International Journal of solids and Structures 47(2010)3214-3220 △K=△K1+i△Kn=//f4(r.0)D(r,0d Evans, AG Heuer, AH 1980. Review-transformation toughening in ceram martensitic transformations in crack-tip stress fields. J. Am. Ceram. So Fischer, F D, Boehm, H-, 2005. On the role of the transformation eigenstrain in the 3 growth or shrinkage of spheroidal isotropic precipitations. Acta Mater. 53, 367- It can be seen from this example that the proposed formulation Gupta, TK Bechtold, JH, Kuznicki, RC, Cadoff, LH, Rossing. B.R. 1977 is also efficient for an infinite transformation area. while the shape of the transformation zone is particularly simple in this case, in Hannink, RH.J 1978. Growth morphology of the tetragonal phase in partially principle the formulation can be used for more complex zone stabilized zirconia. Mater Sci. 13, 2487-245 shapes, e. g of discretization is employed. B.C. 2000. Transformation toughening in 6.3. Example 3: ferroelastic transformation of a circular area enclosing a crack tip Hwang. S.C., Lynch, C.S., McMeeking, R.M., 1995. Ferroelectric/ferroelastic interactions and a polarization switching model. Acta Metall. M 2073-2084 We revisit the transformation problem shown in Fig. 5 We con sid er the intertwining domain distribution with orientation angles of 45 and-45, respectively, within a circular area(see Fig. 5(a)). Jones, JL, Salz, C.R. Hoffman, M, 2005. Ferroelastic fatigue of a soft PZT ceramic. J. uppose that only the domains with the orientation angle -45 in Jones, ,L, Motahari, S M, varlioglu, M, Lienert, U, Bernier. J.V. Hoffman, M the upper circular area Au suffer a 90 -switching after loading. domain switching in a soft lead while the domains with orientation angle 45 in the lower circular zirconate titanate ceramic. Acta mater. 55. 5538-5548. area A, suffer 90 switching, as shown in Fig. 5(b). The radius of the related topics. North-Holland Series in Applied Mathematics and Mechanics, get 6. Effect of nucleation on transformation toughening. J.Am. AK=AK)+iAK =//s(r. 4)D(r, da toughness.J Mater. Sci. 17,235-23g ening-part 2, contribution to fracture Li, ZH, Yang, LH, 2002. The application of the Eshelby equivalent inclusion metho +/5(c:pou- (6.6) Love. AEh. 1927. Mat cal Theory of Elasticity. Cambridge University Press. It can be seen from(6.6)that AKm=0 and a mode l opening crack is Ma, L. Lu, TJ,Korsunsky, A.M., 2006. Vector J-integral analysis of crack interaction with pre-existing singularities. J Appl. Mech. 73, 876-883 Magnani, G, Brillante, A, 2005. Effect of the comp and sintering process on The proposed formulation is thus demonstrated to be also effi- cient for ferroelastic transformation toughening problems with Eur Ceram. Soc. 25. 3383-3392. partial transformation. brittle materials. J Am. Ceram Soc. 65, 242-246. Munz, D Fett, T, 1998. Ceram ties failure behaviour material 7. Conclusions Muskhelishvili, N.L., 1953. Some problems of mathematical theory of ela The fundamental solutions have been obtained for a trans- Pojprapai, S, Jones, J.L, Studer, AJ, Russell.]. Valanoor, N Hoffman, M formed strain nucleus located in an infinite plane and in a plane rroelastic domain switching fatigue in lead zirconate titanate ceramic containing a semi-infinite crack. On the basis of these results, the Rauchs, G Felt, T. Munz, D, Oberacker, R, 2001. Tetragonal-to-monoclinic solutions have been developed and presented in the simple forms ansformation in CeOz-stabilized zirconia under uniaxial loading. J.Eur for toughening induced by the martensitic transformation and the Soc.21.2229-2241. ferroelastic transformation. Finally, some simple examples have been presented to demonstrate that the proposed fundamental for- Ceram Soc. 22.841-849 mulation will pave the way for more rigorous studies of transfor- Rose, LRF- 1987. The mechanics of transformation toughening. Proc R Soc. Lond. 412,169-19 Suo, Z, 1989. Singularities interacting with interfaces and cracks. Int. Solids Struct. Acknowledgements Tsukamoto, H, Kotousov, A, 2006. Transformation toughening in zirconia-enriched The author thanks Professor Robert M. McMeeking for his stim- Vena, P- Gastaldi, D.Contro, R,Petrini,L, Finite element analysis of the fatigue crack growth rate in transformation toughening ceramics. IntJ Plast. ulation, encouragements, and comments throughout the course of 2.895-920 the work. the author is also indebted to helpful discussions with Wang.J, ShL, $.Q- Chen, LQ, Li, Y, Zhang. TY. 2004. Phase held Professor Alexander M. Korsunsky. This work has been partially ported by the National Basic Research Program of China (Grant Yang. w Zhu, T, 1998. Switch-toughening of ferroelectrics subjected to electric 770-5) Yi,S, Gao, S, 2000. Fracture toughening mechanism of shape memory alloys due to martensite transformation. Int J. Solids Struct. 37, Refer Yi,S Gao, S. Shen, LX alloys under mixed-mode loading due to martensite transformation. Int. J. Solids Zeng. D, Katsube, N, Soboyejo, W0, 1999. Simulation of transformation phase. J. Am. Ceram Soc. 59, 49-51 Clussen,N,Ruehle, M, Heuer, H, 1984. Science and technology of zirconia ll. Zeng. D. Katsube, N Soboyejo, w.O., 2004. Discrete modeling of transformation 12. American Ceramic Society, Columbus, OH. toughening in heterogeneous materials. Mech. Mater. 36, 1057-1071 Evans, A.G. Cannon, R.M. 1986. Toughening of brittle solids by martensitic transformations. Acta Metall. Mater. 34. 761-800
DK ¼ DKI þ iDKII ¼ Z Z A f4ðr; hÞDðr; hÞdA ¼ 8 ffiffiffi 2 p 3 þ 16 ! le0 ð1 þ jÞ ffiffiffi R p r ð6:5Þ It can be seen from this example that the proposed formulation is also efficient for an infinite transformation area. While the shape of the transformation zone is particularly simple in this case, in principle the formulation can be used for more complex zone shapes, e.g. of discretization is employed. 6.3. Example 3: ferroelastic transformation of a circular area enclosing a crack tip We revisit the transformation problem shown in Fig. 5. We consider the intertwining domain distribution with orientation angles of 45 and 45, respectively, within a circular area (see Fig. 5(a)). Suppose that only the domains with the orientation angle 45 in the upper circular area Au suffer a 90-switching after loading, while the domains with orientation angle 45 in the lower circular area Al suffer 90 switching, as shown in Fig. 5(b). The radius of the circular area is R and D(r,h) = 1/2. Substituting the influence function (5.4) into Eq. (5.5), after a straightforward manipulation, we get DK ¼ DKI þ iDKII ¼ Z Z Au f5 r; h; p 4 Dðr; hÞdA þ Z Z Al f5 r; h; p 4 Dðr; hÞdA ¼ 16 7 le0 ffiffiffi R p ffiffiffiffiffiffi 2p p ð1 þ jÞ ð6:6Þ It can be seen from (6.6) that DKII = 0 and a mode I opening crack is obtained. The proposed formulation is thus demonstrated to be also effi- cient for ferroelastic transformation toughening problems with partial transformation. 7. Conclusions The fundamental solutions have been obtained for a transformed strain nucleus located in an infinite plane and in a plane containing a semi-infinite crack. On the basis of these results, the solutions have been developed and presented in the simple forms for toughening induced by the martensitic transformation and the ferroelastic transformation. 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