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:201008002Fundamental 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 demon￾strate 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 con￾trolled 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; Mag￾nani 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 trans￾form from tetragonal to monoclinic, accompanied by a volume increment of 4% and a shear strain of 16%. These strains induce fur￾ther 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 do￾main 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 tough￾ening 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 dis￾crete 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 trans￾formed regions. In this paper, we concentrate on the Green’s func￾tion 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) ob￾tained 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 trans￾formation 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 dilat￾ant 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