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 pres￾ent paper we carry out a thorough and systematic study of the Green’s function method for formulating problems of transforma￾tion 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 solu￾tion, 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 Sec￾tion 5, fundamental formulations for transformation toughening for martensitic and ferroelastic phenomena are developed with the Green’s function method. To demonstrate the validity and rel￾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. 2. Fundamental solution for a transformed strain nucleus located in an infinite plane solid In this section, we seek the Muskhelishvili potentials for a trans￾formed strain nucleus located in an infinite plane solid. The strain nucleus will be modeled in terms of the mathematical edge dislo￾cation 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 func￾tion denotes its complex conjugate. It is known that the Muskhe￾lishvili 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 trans￾formed 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 ori￾entation 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 dis￾locations 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