ScienceDirect. Hamiltonian approach to piezoelectric fracture

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1 Available online at ScienceDirect Procedia Materials Science 3 ( 014 ) th European Conference on Fracture (ECF0) Hamiltonian approach to piezoelectric fracture J.M. Nianga a b, c*, N. Recho a Hautes Etudes d Ingénieur de Lille, 13 rue de Toul, Lille-France b ERMESS, EPF Ecole d ingénieurs, 3 bis, rue Lakanal, 9330 Sceaux-France c Pascal Institute, Blaise Pascal niversity, 63000, Clermont Ferrand-France Abstract The electroelastic behavior of a piezoelectric multi-material with a V-notch is investigated in this study, via a Hamiltonian approach. At first, we consider the piezoelectric field of the structure, by extending ZHONG s formalism [1] to linear piezoelectricity, through the Two Extreme Point s problem. Following a unified description of ZHONG and BI s formalisms [], we present a formulation of electro elastic fields at the notch tip, through the canonical equations of Hamilton. A piezoelectric multi-material problem is thus reduced to a single-material problem, with a relative simplification of the constitutive equations in the form of a first-order ordinary differential system, from which derives an original method for determining the singularity degree for two-dimensional piezoelectric structures with a V-notch. The procedure is then applied to a bi-material piezoelectric specimen in order to determine the stress amplitude and the singularity degree close to the notch tip as function of the electric field and the notch angle. 014 Elsevier Ltd. Open access under CC BY-NC-ND license. 014 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the Norwegian niversity of Science and Technology (NTN), Department Selection and peer-review under responsibility of the Norwegian niversity of Science and Technology (NTN), Department of of Structural Engineering Structural Engineering. Keywords: V-notch singularity,hamiltonian approach, piezoelectric fracture, eigenvalue problems * Corresponding author. Tel.: +33 (0) ; fax: +33 (0) address: naman.recho@epf.fr Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the Norwegian niversity of Science and Technology (NTN), Department of Structural Engineering doi: /j.mspro

2 J.M. Nianga and N. Recho / Procedia Materials Science 3 ( 014 ) Model and methodology In this paper, the model and the resolution methodology are established in the case of -D medium of a linear elastic material containing a V-notch with polar coordinates (fig.1). The obtained solution will be extended to the case of a bi-material specimen (Fig.) representing a joining dissimilar piezoelectric materials specimen. θ O ρ θ 1 Fig.1: V-notch specimen with polar coordinates Fig.. V-notch for a bi-material 1.1. The eigenvalues problem of ordinary differential equations for the V-notches in the field of linear piezoelectricity In this section, an original method of determination of singularity degrees for piezoelectric two-dimensional structures presenting a V- notch is proposed. At first, the fundamental equations of linear piezoelectricity are reduced to a problem of finding eigenvalues, through the resolution of ordinary differential equations admitting as unknown, the angular coordinate θ around the notch tip. We thus apply to piezoelectricity, the procedure allowing determining the singularity degrees as recently developed in []. For that, we use the interpolation matrix method, in order to solve the corresponding differential equations. Therefore, the eigenvectors at the vicinity of the notch tip, respectively associated to the electric field E, to the electric potential φ and to the electric and mechanical displacements D and u, are then determined. Now, let consider a V-notched piezoelectric specimen with an opening angle 1 (Fig.1). The notch tip being chosen as origin of the polar coordinate system (ρ,θ), the electromechanical fields in its vicinity can then be formulated as an asymptotic expansion, according to the radial coordinate ρ [3] 1 u(,) () 1 u (, ) ( ) 1 (, ) ( ) (1) The real number λ is an eigenvalue, whereas p (θ), θ (θ) and Ψ(θ) respectively correspond to the eigenvectors. Combining Equations (1) with one hand the conventional mechanical relationship "displacement-strain", and secondly, with those between the electric field and electric potential, it respectively follows, for both the components of the strain tensor γ and the electric field E, when introducing the following notation. d d.

3 30 J.M. Nianga and N. Recho / Procedia Materials Science 3 ( 014 ) (, ) ( 1) ( ); E (, ) ( 1) ( ) (, ) ( ) ( ); E (, ) ( ) (, ) ( ) ( ) () Moreover, for a two-dimensional piezoelectric material with hexagonal symmetry, the generalized Hookes law can be formulated as follows c c e E ; D e e E c1 c e31e e15e ; D e15 11E c11 c1 e15e (3) We respectively denote by σ, E, γ and D, the stress field, the electric field, the strain field, and the electric displacement field. All these fields are expressed in a polar coordinate system. The tensor of elastic coefficients, the tensor of piezoelectric coefficients and that of dielectric coefficients are respectively denoted by C, e and ε. Substituting () into the piezoelectric constitutive equations it then follows, for all 1, c11 c1 ( ) e 15 ( ) c11 c1 c ( ) e 15 ( ) c11 1 c ( ) 1e33 1e 31 ( ) 0 c ( ) e 15 ( ) c11c c11 c1 ( ) e15 e31 1 ( ) c11 c1 ( ) 0 e15 ( ) 11 ( ) e31 e15 e31 ( ) 33 1 ( ) e33 1e31 ( ) 0 (4) Assuming first, that tractions on the notch lips and near its tip are zero, and secondly, that the dielectric permittivity in the space between those lips also equals zero, we can then respectively write 0 ; D 1 D (5) Subsequently, by substituting these equations into the relationship (), it follows, for 1 and e 15 ( ) ( ) 11 ( ) 0 c1 1 c ( ) c ( ) e31 1 ( ) e 15 ( ) 0 c11 c ( ) ( ) e 15 ( ) 0 (6)

4 J.M. Nianga and N. Recho / Procedia Materials Science 3 ( 014 ) The linearity of Equations (4) with respect to the eigenvalue λ may then be obtained, for 1,, through the introduction of the following auxiliary functions g ( ) ( ); g ( ) ( ); g( ) ( ) (7) Moreover, these equations can be transformed as follows c11 c c c ( ) e ( ) c c c ( ) e ( ) c g ( ) c c c ( ) e g( ) e e e e ( ) 0 c ( ) e ( ) c c c c ( ) e e 1 ( ) g() 0 e ( ) ( ) e e e ( ) g( ) 1 ( ) e33 1 e31 ( ) 0 (8) Evaluation of singularity degree in the vicinity of the notch tip, is consequently reduced to solving a linear problem of determining eigenvalues.this is governed by Equations (7) and (8) to which are added the boundary conditions (5). The determination of the eigenfunctions should then allow that of stresses and electric displacements in the vicinity of the notch tip. 1.. Determination of singularity degree of V-notches for joining dissimilar piezoelectric materials Consider a medium composed of two different piezoelectric materials forming a notch (fig.). Assigning, respectively, the latter by the subscripts 1 and, we can rewrite Equations (7) and (8), as follows (i) (i) (i) (i) (i) (i) g ( ) ( ); g ( ) ( ); g ( ) ( ) (i) (i) (i) (i) (i) (i) (i) (i) (i) c11 c1 ( ) e 15 ( ) c11 c1 c ( ) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) e 15 ( ) c11 g ( ) c11 c11 c ( ) e33g ( ) (i) (i) (i) (i) (i) (i) (i) (i) (i) c ( ) e 15 ( ) c11 c c11 c1 ( ) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) e15 ( ) 11 ( ) e31 e15 e 31 ( ) 33 g ( ) (i) (i) (i) (i) (i) e e e e ( ) 0 (i) (i) (i) (i) (i) (i) e e 1 ( ) c c g ( ) 0 (i) (i) (i) (i) (i) 33 1 ( ) e33 1e 31 ( ) 0 1, pour i 1 et 3, 4 pour i (9)

5 3 J.M. Nianga and N. Recho / Procedia Materials Science 3 ( 014 ) Note that ( i ) ( i, ) and ) p (i, respectively represent the components of the displacement and that of ( i) electric field, in the sub-field i 1,, in the vicinity of the notch tip. However, at the interface between the two materials, the conditions of continuity of the components of the displacement, as well as, those of electric flow on the one hand, those corresponding to the components of the stress tensor field and those of electric displacement on the other hand, lead to the following relationship (1) () (1) () ; (1) () (1) () (1) () D D ; (1) () (1) () D D (10) Comparing equations () - (3) and (10), it follows, for (1) (1) (1) (1) (1) (i) (1) (1) (1) c1 1 c ( ) c ( ) e31 1 ( ) e 15 ( ) () () () () () () () () () c1 1 c ( ) c ( ) e31 1 ( ) e 15 ( ) (1) (1) (1) (1) (1) (1) c11 c1 ( ) ( ) e 15 ( ) () () () () () () c11 c1 ( ) ( ) e 15 ( ) (1) (1) (1) (1) (1) (1) e15 ( ) e15 ( ) 11 ( ) () () () () () () e15 ( ) e15 ( ) 11 ( ) (11) The notch lips being assumed free of stress and free of electric displacement, the following boundary conditions are respectively needed for 1 and 3 (1) (1) (1) (1) (1) (i) (1) (1) (1) c1 1 c ( ) c ( ) e31 1 ( ) e 15 ( ) 0 (1) (1) (1) (1) (1) (1) c11 c1 ( ) ( ) e 15 ( ) 0 (1) (1) (1) (1) (1) (1) e15 ( ) e15 ( ) 11 ( ) 0 ( 1 ) (1) () () () () () () () () () c1 1 c ( ) c ( ) e31 1 ( ) e 15 ( ) 0 () () () () () () c11 c1 ( ) ( ) e 15 ( ) 0 () () () () () () e15 ( ) e15 ( ) 11 ( ) 0 ( 3 ) (13) The problem of determining the singularity degree in the vicinity of the notch tip resulting from assembly of two dissimilar piezoelectric materials is reduced in a simple system of ordinary differential equations (9), which admits as boundary conditions, Equations (8) - (13).

6 J.M. Nianga and N. Recho / Procedia Materials Science 3 ( 014 ) Resolution Procedure: Matrix method of interpolation for solving eigenvalue problems for ordinary differential equations form To solve Equations (9), we propose to reformulate them, for 1, g ikj( )y k ( ) q ikj( )y k ( ) 0 (i 1,...,6; 1, ) k1j0 k1j0 j 0,...,m; i mi for k, i 1, 1,,, 3, 3 j0, mi 0fork,i4,4,5,5,6,6, in the following generic and simplified ; r = 6 (14) Where the real numbers g ijk and q ijk depend on the angular variable θ and where the variables k ; m 1, y m k are defined as follows y ;y ;y ;y g ;y g ;y g;y ;y y ;y g ;y g ;y g;y ;y ;y ;y g () () y5 g ;y6 g (0) (0) (0) (0) (0) (0) (1) (1) (1) (1) (1) (1) () () () () (15) Applying the interpolation matrix method, Equation (14) can then be formulated for each discretization node as follows G Y Gikj Yk Qikj Yk k1 j0 k1 j0 ( ) ( ) ( ) ( ) 0 (i 1,...,6) diag[g (x ),g (x ),...,g (x )]; G diag[q (x ),q (x ),...,q (x )] ikj ikj 0 ikj 1 ikj n ikj ikj 0 ikj 1 ikj n ( ) y (x ), y (x ),..., y (x ) k k 0 k 1 k n (j1) (j1) Yk ( ) y k (x 0) σ DYk ( ) R k 1; T (16) D and R respectively denote a matrix by means of the Lagrangian polynomial and a residual errors vector. Furthermore, the eigenvalues problem is then reduced to the following equations whose the solution is easier to get 1 Y0 Y0 CC 1 Y Y T (m 1) (m ) (m r) 0 Y10 Y 0...Y r0 ; Y1 Y...Y Y Y r The matrices C 1 and C are formulated on the basis of electromechanical coefficients. So, once determined the eigenvalues and eigenvectors associated with this former equation, the eigenvectors corresponding to the derivatives m Y k 1 k r k :1.., can be then calculated with the following equation T (17)

7 34 J.M. Nianga and N. Recho / Procedia Materials Science 3 ( 014 ) Yk Pkj k0d Yk k mk 1 j kj P σ Dσ D σ mkj (m k) Y k:1...r; j:1...m 1 (18) 3. Future extensions The procedure developed here can be applied to any D-elastic piezoelectric medium. In order to extend the use of this procedure to any specimen submitted to piezoelectric field, a V-notch box, as in figure 1, will be programmed as function of a given radius and given boundary conditions. A displacement field, issued from previous finite element analysis, represents these boundary conditions. Numerical parametric calculations will be done, in terms of J-Integral, in order to choice the suitable radius to be considered in connecting finite element analysis to local analysis. The comparison with experimental results issued from the bibliography will be done in the near future. 4. Conclusion. It is obvious that the accuracy of the used method depends on the degree of implemented interpolation polynomials. For this purpose, a modelling by spline functions is of some interest. This method has moreover a significant advantage in obtaining, with the same reliability, derivatives of all orders, appearing in boundary value problems governed by ordinary differential equations. The calculation of piezoelectric stress and electric fields that require evaluation of the first derivative, both for the mechanical displacement and the electric potential, is therefore particularly suitable. References 1. W. X. Zhong, (1995). A new systematical methodology in elasticity theory (in Chinese). Dalian Science & Technology niversity Press.. J. Li, N. Recho (00). Méthodes asymptotiques en mécanique de la rupture (in French). Editions Hermes Sciences, Paris. 3. H. A. Sosa and Y. E. Pak coordinated, Three dimensional eigenfunctions analysis of a crack in apiezoelectric material. Int. J. Solids. Struct. 6, 1, C. M. Kuo and D. M. Barnett, 1991 Stress singularities of interfacial cracks in bonded piezoelectric half-spaces. In Modern Theory of Anisotropic Elasticity and Applications, Eds. W. J. J. Ting T. C. T., and Barnett D. M, SIAM. Pp Z. Suo., C.-M. Kuo., D. M. Barnett., J. R. Willis., 199. Fracture Mechanics forpiezoelectric ceramics. J. Mech. Phys. Solids, vol. 40, no. 4, pp Z. Niu., D. Ge., C. Cheng., J. Ye, N. Recho, 009. Evaluation of the stress singularities of plane V-notches in bonded dissimilar materials, Applied Mathematical Modelling, vol. 33, no. 3, pp Z. Niu., C. Cheng., J. Ye, and N. Recho, 009. A new boundary element approach of modelling singular stress fields of plane V-notch problems. International Journal of Solids and Structures, vol. 46, no. 16, pp