interaction energy of interface dislocation loops in piezoelectric bi crystals

Accepted Manuscript Interaction energy of interface dislocation loops in piezoelectric bi-crystals Jianghong Yuan, Yin Huang, Weiqiu Chen, Ernian Pan PII: DOI: Reference: S2095-0349(17)30011-9 http://dx.doi.org/10.1016/j.taml.2017.02.002 TAML 127 To appear in: Theoretical & Applied Mechanics Letters Received date: 28 November 2016 Accepted date: 14 February 2017 Please cite this article as: J Yuan, Y Huang, W Chen, E Pan, Interaction energy of interface dislocation loops in piezoelectric bi-crystals, Theoretical & Applied Mechanics Letters (2017), http://dx.doi.org/10.1016/j.taml.2017.02.002 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain Interaction energy of interface dislocation loops in piezoelectric bi-crystals Jianghong Yuan1,2,†,*, Yin Huang3,†, Weiqiu Chen4,5, Ernian Pan6 State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, Sichuan 610031, China School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China Center for Mechanics and Materials, and AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou 310027, China Department of Civil Engineering, and Department of Applied Mathematics, University of Akron, Akron, OH 44325, USA Abstract: Interface dislocations may dramatically change the electric properties, such as polarization, of the piezoelectric crystals In this paper, we study the linear interactions of two interface dislocation loops with arbitrary shape in generally anisotropic piezoelectric bi-crystals A simple formula for calculating the interaction energy of the interface dislocation loops is derived and given by a double line integral along two closed dislocation curves Particularly, interactions between two straight segments of the interface dislocations are solved analytically, which can be applied to approximate any curved loop so that an analytical solution can be also achieved Numerical results show the influence of the bi-crystal interface as well as the material orientation on the interaction of interface dislocation loops Keywords: interface dislocation, piezoelectric bi-crystals, anisotropic elasticity, interaction energy Introduction Interface dislocations may severely degrade ferroelectric properties around the dislocation core owing to couplings between the polarization and the stresses induced by dislocations [1] Since the piezoelectric effect never exists in isotropic materials, the factor of anisotropy must be taken into account when dealing with the interaction between dislocations in piezoelectric crystals Although there are some elegant works † * These authors contributed equally to this work Corresponding author, E-mail address: yuanjianghong1984@gmail.com in the literature investigating the displacements and stresses of dislocation loops within the anisotropic half-spaces [2-4], bi-materials [5-11] and multilayered solids [12-14], no explicit line-integral formula for calculating the interaction energy of dislocation loops, which involves both the interface effect and the material anisotropy, is available to the authors’ knowledge In this work, we derive a simple formula for calculating the interaction energies of planar interface dislocation loops with arbitrary shape in the piezoelectric bi-crystals The advantage of our formula lies in the fact that it is given by a double line integral along two dislocation curves, with the simplicity similar to the corresponding expression in the anisotropic elastic full-space [15] Presented in this paper are also the analytical expressions of the interaction energy due to two straight segments of interface dislocations so that a curved dislocation loop can be approximated in a piecewise manner This paper is outlined below The problem to be solved is described in Sect As a preliminary, the Green’s functions in the generally anisotropic piezoelectric bi-crystals are summarized in Sect The line-integral formula for calculating the interaction energy of the interface dislocation loops within the anisotropic piezoelectric bi-crystals is derived in Sect 4, where the interaction between two straight dislocation segments is also discussed in detail Numerical results are presented in Sect 5, which clearly show the effects of bi-crystal interface and material orientation on the interaction energy between interface dislocations A concluding remark is drawn in Sect as the closure of this paper Problem description Throughout this paper, the Einstein’s summation convention for repeated indices will be used, with the Greek indices (such as , ) running from to 2, the lowercase Roman ones (such as j, k) from to while the uppercase Roman ones (such as J, K) from to The bi-crystal model considered here is composed of two anisotropic piezoelectric half-spaces perfectly bonded on the interface plane x3  , as is shown in Fig The half-space #1 ( x3  ) is (1) occupied by material #1 with the extended elastic coefficients ciJKl , and the half-space #2 ( x3  ) is (2) ( ) occupied by material #2 with ciJKl According to the Barnett-Lothe notation [16], ciJKl is given by ( ) ( ) ciJKl  clKJi ( ) ) ( )  cijkl ,  c (jikl  cklij  ( ) ( )  eikl  eilk ,   ( ) ( )  elij  elji ,  ( ) ( )   il   li , J  j  1, 2,3, K  k  1, 2,3, J  4, K  k  1, 2,3, J  j  1, 2,3, K  4, (1) J  4, K  4, in which  ij , eijk and cijkl are the dielectric, piezoelectric and elastic coefficients, respectively In order to solve the interaction between two planar dislocation loops located on the interface of the aforementioned bi-crystal, we first consider two planar, arbitrarily shaped dislocation loops that are both parallel to but away from the bi-crystal interface (Fig 1) The first loop C1 , located within the half-space #1, has a constant extended Burgers vector b1 over the planar cut-surface A1 ; and the second loop C2 , located in the half-space #2, has a constant extended Burgers vector b2 over the planar cut-surface A2 The above extended Burgers vector actually has components in column: the first three components define a classical dislocation with a displacement discontinuity across the cut-surface, and the fourth component defines an electric dipole layer characterized by a jump in the electrical potential across the cut-surface [16] In the next sections, we will first deduce a line-integral formula for calculating the interaction energy between loops C1 and C2 , and then obtain the desired interaction energy expression for interface dislocation loops by moving the two loops to the interface from both sides Fig.1 The bi-crystal model and the coordinate system attached to it Two planar dislocation loops in the bi-crystal are both parallel to but away from the interface The special interface dislocation case can be obtained by moving the two loops to the interface from both sides of the interface Green’s function for anisotropic piezoelectric bi-crystals We now define two fundamental 44 matrices u(1) ( x; y) and t (1) ( x; y) The (I, J) element of u(1) ( x; y) and t (1) ( x; y) denotes the I-component of the extended displacement and the 3I-component of the extended stress at point x ( x1 , x2 , x3 ) in the half-space #1, respectively, due to the unit J-component of the extended body-force at point y ( y1 , y2 , y3 ) in the half-space #2 The abovementioned extended displacement has components: the first three denote the elastic displacements while the fourth denotes the electric potential; the extended body-force also has components: the first three are the mechanical body-forces and the fourth is the negative electric charge density, respectively; the extended stress has columns: the first three are the mechanical stresses and the fourth is the electric displacement, respectively The matrix u(1) ( x; y) can be expressed compactly as Ref [17,18] u(1)  x; y   2π  π d Re  A(1)G  x, y,   A(2)T  , (2) where Re{ } denotes the real part of complex numbers, and G  x , y,   is a 44 matrix whose elements are defined by  S IJ G  x, y,   IJ  f IJ  x, y,   (3) In Eq (3), S is a 44 matrix defined by S  A(1)1  M (1)  M (2)  1 M (2)  M (2)  A(2) , (4) which depends only on the material properties of the bi-crystal Here, M (  )  iB(  ) A(  )1 is called the extended impedance tensor of material #, the overbar denotes the complex conjugate, and “i” is the imaginary unit The function f IJ  x , y,   in Eq (3) is defined by f IJ  x, y,    n  x  y   pI(1) x3  pJ(2) y3 , with n1  cos  and (5) n2  sin  We also point out that A(  )  {a1(  ) , a2(  ) , a3(  ) , a4(  ) } and B(  )  {b1(  ) , b2(  ) , b3(  ) , b4(  ) } introduced in Eqs (2) and (4) are the 44 eigen-matrices of material # in Stroh formalism corresponding to the eigen-value vector { p1(  ) , p2(  ) , p3(  ) , p4(  ) } appearing in Eq (5), which satisfy the normalization relation below B(  )T A(  )  A(  )T B(  )  I 44 , (6) with I44 being the 44 identity matrix More specifically, the above aI(  ) and bI(  ) denote the I-th eigen-vectors of material # associated with the eigen-value pI(  ) , and they are related by bI(  )    pI(  )1Q (  )  R(  )  aI(  )   R(  )T  pI(  )T (  )  aI(  ) (7) Note that pI(  ) has a positive imaginary part and is the I-th eigen-value of the eigen-relation below related to material #: Q (  )  pI(  )  R(  )  R(  )T   pI(  )2T (  )  aI(  )  ,   (8) in which T (  ) , R(  ) and Q (  ) are all the 44 matrices whose elements are defined by ( ) ( ) ( ) [T (  ) ]JK  ciJKl mi ml , [ R(  ) ]JK  ciJKl ni ml , and [Q (  ) ]JK  ciJKl ni nl , (9) respectively, with n1 , n2 , n3 T  cos  ,sin  ,0 T and m1 , m2 , m3 T  0,0,1 T (10) Further, t (1)  x; y  can be easily obtained from Eq (2) as t (1)  x; y   2π  π d n  Re B(1)G  x, y,   A(2)T  x (11) Equation (11) will be particularly useful in the following section Interaction energy between two dislocations in anisotropic piezoelectric bi-crystals 4.1 Line-integral expression for the interaction energy We now define two 41 vectors u(2) ( y; C1 ) and t (2) ( y; C1 ) The J-th component of u(2) ( y; C1 ) and t (2) ( y; C1 ) means the J-th component of the extended displacement vector and the 3J-th component of the extended stress tensor at point y ( y1 , y2 , y3 ) in the half-space #2, respectively, due to the planar dislocation loop C1 that is located in the half-space #1 and parallel to the bi-crystal interface According to the theory of dislocations, u(2) ( y; C1 ) can be expressed as Ref [19] u(2)  y; C1     dA  x  t (1)T  x; y  b1 , (12) A1 in which dA  x  is the area element at point x on the cut-surface A1 By substituting Eq (11) into Eq (12) and then using the Green’s formula below  A  f f1     dA    C f dx ,  x1 x2  (13) the area integral over A1 in Eq (12) can be easily transformed to a line integral along C1 as 2π u(2)  y; C1     dx  l d Re  A(2)G T  x, y,   B(1)T b1 , π C1 (14) with l1  n2 and l2  n1 Furthermore, t (2)  y;C1  can be derived from Eq (14) as t (2)  y; C1    2π  C1 π dx  l d n  Re B(2)G T  x , y,   B(1)T b1 y  (15) Let W (C2 ; C1 ) denote the interaction energy between the aforementioned dislocation loops C1 and C2 (Fig 1) According to the theory of dislocations [19], we have W (C2 ; C1 )   dA  y  t (2) T  y; C1  b2 , (16) A2 in which dA  y  is the area element at point y on the cut-surface A2 Substituting Eq (15) into Eq (16), along with one more application of Eq (13), finally transforms Eq (16) to the double line integrals along C1 and C2 as follows: W (C2 ; C1 )   2π  C1 dx  dy  l l d Re b1T B (1)G  x, y,   B(2)T b2  π C2 (17) 4.2 Interaction energy between straight dislocation segments If the aforementioned dislocation loops C1 and C2 are composed of some straight segments, then the interaction energy between them, as derived in Eq (17), can be analytically obtained by summing the interaction energies between the individual segments [19] Let us consider one straight segment on C1 from point x sta ( x1sta , x2sta , x3 ) to x end ( x1end , x2end , x3 ) and another on C2 from point ysta ( y1sta , y2sta , y3 ) to yend ( y1end , y2end , y3 ) , the parametric equations of which are written, respectively, as x  xsta  t1  xend  xsta  ,  t1  1, (18) y   y sta  t2  yend  ysta  ,  t2  Actually, the double line integral as shown in Eq (17) can be integrated analytically along the above straight dislocation segments by observing the following relation:  x end x sta  dx  yend ysta dy f IJ1  x , y,     xend  xsta  yend  ysta   f IJ  x end , yend ,   ln f IJ  x end , yend ,    f IJ  x sta , yend ,   ln f IJ  x sta , yend ,       f IJ  x end , ysta ,   ln f IJ  x end , ysta ,    f IJ  x sta , ysta ,   ln f IJ  x sta , ysta ,       f IJ  x end , yend ,    f IJ  x sta , yend ,    f IJ  x end , yend ,    f IJ  x end , ysta ,      (19) 4.3 Interface dislocation loops If the dislocation segments are located on the bi-crystal interface, then f IJ1  x , y,   tends to infinity once ˆ  n  x  y  vanishes Thus it becomes necessary to take the limits on both sides of Eq (19) as follows: lim x3 0 , y3 0 lim  x3 0 , y3 0 ln f IJ  x , y,    ln ˆ  iπH   ˆ  , (20) f IJ1  x, y,   ˆ 1  iπ  ˆ  in which H  z  is the so-called Heaviside step function in z and   z  is the Dirac delta function in z In so doing, the analytical expression on the right-hand side of Eq (19) actually behaves as a regular function in , provided that any three points among x sta , x end , ysta and yend are non-collinear However, if at least three points among them are collinear, then singularity occurs at a particular value of  and we have to remove it by evaluating the so-called finite-part integral in  The treatment of the finite-part integral is quite similar to that used in Ref [20] and thus omitted here In summary, the expression of the interaction energy between two interface straight dislocation segments is generally given by the regular integrals in   [0, π] , and we can thus calculate it accurately by the Gauss-Legendre quadrature If the dislocation loop is made of curved lines, we can approximate them piecewise using the straight line segments Therefore, the interaction energy of interface dislocation loops with arbitrary shape is finally solved in terms of simple analytical expression Numerical Example and discussion The example considered here involves a bi-crystal composed of two perfectly bonded BaTiO3 half-spaces with dissimilar material orientations The symmetry axes (i.e poling directions) of half-space #1 and half-space #2 are along the vectors { sin(  ),0,cos(  )}T and { sin  ,0,cos  }T , respectively, in which  (  0) is the counterclockwise rotation angle about the x2-axis The elastic, piezoelectric and dielectric constants of the transversely isotropic BaTiO3 in its material coordinates are listed below [21]: {c11 , c12 , c13 , c33 , c44 }  {166,77,78,162, 43}GPa , and {e31 , e33 , e15}  {4.4,18.6,11.6}C/ m , and {11 ,  33}  {11.2,12.6}  109 C2 / (N  m ) Two square interface dislocation loops, both having the same extended Burgers vector {b,0,0,0}T, are investigated here The first square loop has a side length of L, with its centroid located at point (0,0,0) and each side being parallel/perpendicular to the coordinate axes The second loop is given by a simple translation of the first one, with its centroid located at point ( xc , yc ,0) Figure shows the contours of the normalized interaction energy W / ( c44b2 L) on the ( xc / L, yc / L) -plane for rotation angle   45 , in which the bi-crystal solution is compared with the full-space solution corresponding to material #1 or #2 [6, 22] Note that the full-space solution for material #1 is identical to that for material #2 in this example We can observe an obvious difference in Fig between the bi-crystal solution and the full-space solution Fig The contours of the normalized interaction energy W / ( c44b2 L) on the ( xc / L, yc / L) -plane for rotation angle   45 Figure shows the curves of the normalized interaction energy varying with the rotation angle  for ( xc / L, yc / L)  (1.5,1.5) , in which the bi-crystal solution is also compared with the corresponding full-space solution for material #1 or #2 It can be observed that, in the presence of interface, the material orientation has a significant influence on the interaction energy as compared to the case without interface Fig Normalized interaction energy W / ( c44b2 L) vs rotation angle  for ( xc / L, yc / L)  (1.5,1.5) Concluding remarks We have presented in this paper an explicit line-integral formula for calculating the interaction energy of arbitrarily shaped interface dislocation loops in the generally anisotropic piezoelectric bi-crystals It is shown from numerical calculations that the presence of interface along with the material orientation has an obvious influence on the interaction energy of two interface dislocation loops This work could be considered as a theoretical contribution to the theory of dislocations Acknowledgments J.H.Y and Y H acknowledge the supports from the National Natural Science Foundation of China (11402133 and 11502128) References [1] S.P Alpay, I.B Misirlioglu, V Nagarajan, et al., Can interface dislocations degrade ferroelectric properties? 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