Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering
doi: 10.1016/j.mspro.2014.06.055
ScienceDirect
20th European Conference on Fracture (ECF20)
Hamiltonian approach to piezoelectric fracture
J.M. Niangaa , N. Recho
b, c*
a Hautes Etudes d’Ingénieur de Lille, 13 rue de Toul, 59800 Lille-France b ERMESS, EPF – Ecole d’ingénieurs, 3 bis, rue Lakanal, 92330 Sceaux-France
c Pascal Institute, Blaise Pascal University, 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 BUI’s formalisms
[2], 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.
319 J.M. Nianga and N. Recho / Procedia Materials Science 3 ( 2014 ) 318 – 324
1. Model and methodology
In this paper, the model and the resolution methodology are established in the case of 2-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.2) representing a joining dissimilar piezoelectric materials specimen.
Fig.1: V-notch specimen with polar coordinates Fig. 2. 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 [2]. 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 212 ssr // (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
1
1
u ( , ) U ( )( , ) ( )
u ( , ) U ( )
n-t t n-n-s s
Ê t s ? t sÍ l t s ? t [ sË t s ? t sÍÌ (1)
The real number λ is an eigenvalue, whereas Up(θ), Uθ(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 * + * +..' sd
d?
ρ θ1
θ2
O
320 J.M. Nianga and N. Recho / Procedia Materials Science 3 ( 2014 ) 318 – 324
' '
'
( , ) ( 1) U ( ); E ( , ) ( 1) ( )
( , ) U ( ) U ( ); E ( , ) ( )
( , ) U ( ) U ( )
n ntt t t
n n nss t s s
n nts t s
i t s ? n - t s t s ? / n - t [ si t s ? t s -t s t s ? /t [ si t s ? t s - nt s
ÊÍÍËÍÍÌ (2)
Moreover, for a two-dimensional piezoelectric material with hexagonal symmetry, the generalized Hooke's law
can be formulated as follows
* +11 12 33 33 31 33
12 22 31 15 15 11
11 12 15
c c e E ; D e e E
c c e E e E ; D 2e E
c c e E
tt tt ss t t tt ss tss tt ss t s s ts sts ts s
Êu ? i - i / ? i - i - gÍÍu ? i - i / / ? i - gËÍv ? / i /ÍÌ (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 (2) into the piezoelectric constitutive equations it then follows, for all * +21,sssŒ
* + * +* + * + * +
* + * +* +* +* +
'' '' ' '
11 12 15 11 12 22 15
2
11 22 33 31
'' '' ' '
22 15 11 22 11 12 15 31
11 12
''
15 11
c c U ( ) e ( ) c c c U ( ) e ( )
c 1 c U ( ) 1 e 1 e ( ) 0
c U ( ) e ( ) c c 2c c U ( ) e e 1 ( )
2 c c U ( ) 0
2e U ( )
t s
t
s ts
t
Ç ×/ s - [ s - n - / s - [ sÉ ÚÇ × Ç ×- n - / s - n - n - - [ s ?É ÚÉ ÚÇ ×s - [ s - n - - / s - - n - [ sÉ Ú
- n n - / s ?s / g [ * + * +
* +2'' '
31 15 31 33
33 31
( ) e 2e e U ( ) 1 ( )
e 1 e U ( ) 0
s
t
ÊÍÍÍÍÍËÍÍ Ç ×s - - n - s / g n - [ sÍ É ÚÍ Ç ×- n - - s ?Í É ÚÌ
(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
1 2
1 2
0; D D 0
0
ss ss - /s sts tss?s s?s
u uÃ Ô Ã Ô Ã Ô? ? ? ?Ä Õ Ä Õ Ä Õu u Å ÖÅ Ö Å Ö (5)
Subsequently, by substituting these equations into the relationship (2), it follows, for 1ss ? and 2ss ?
* + * +* +
' '
12 22 22 31 15
' '
11 22 15
' '
15 11
c 1 c U ( ) c U ( ) e 1 ( ) e ( ) 0
c c U ( ) U ( ) e ( ) 0
2e U ( ) U ( ) ( ) 0
t s
t s
t s
ÊÇ ×n - - s - s - n - [ s - [ s ?É ÚÍÍ Ç ×/ s - n s - [ s ?Ë É ÚÍ Ç ×s - n s / g [ s ?Í É ÚÌ (6)
321 J.M. Nianga and N. Recho / Procedia Materials Science 3 ( 2014 ) 318 – 324
The linearity of Equations (4) with respect to the eigenvalue λ may then be obtained, for * +21,sss Œ , through the
introduction of the following auxiliary functions
g ( ) U ( ); g ( ) U ( ); g( ) ( )t t s ss ? n s s ? n s s ? n[ s (7)
Moreover, these equations can be transformed as follows
* + * +* + * + * +
* + * +* +* +
'' '' ' '
11 12 15 11 12 22 15 11
11 11 22 33 33 31 33 31
'' '' ' '
22 15 11 22 11 12 15 31
11 1
c c U ( ) e ( ) c c c U ( ) e ( ) c g ( )
2 c c c U ( ) e g( ) 2e e e e ( ) 0
c U ( ) e ( ) c c 2c c U ( ) e e 1 ( )
2 c c
t s t
t
s t
Ç ×/ s - [ s - n - / s - [ s - n sÉ ÚÇ × Ç ×- n - / s - n s - n - - - [ s ?É Ú É Ú
Ç ×s - [ s - n - - / s - - n - [ sÉ Ú- n - /* +
* + * +* +
2
'' '' '
15 11 31 15 31 33 33
33 31
g ( ) 0
2e U ( ) ( ) e 2e e U ( ) g( ) 2 1 ( )
e 1 e U ( ) 0
st s
t
ÊÍÍÍÍÍË s ?ÍÍ Ç ×s / g [ s - - n - s / g n s / g n - [ sÍ É ÚÍ Ç ×- n - - s ?Í É ÚÌ
(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.
1.2. 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.2). Assigning,
respectively, the latter by the subscripts 1 and 2, we can rewrite Equations (7) and (8), as follows
* + * +* +
(i) (i) (i) (i) (i) (i)
(i) (i) ''(i) (i) ''(i) (i) (i) (i) '(i)
11 12 15 11 12 22
(i) '(i) (i) (i) (i) (i) (i) (i)
15 11 11 11 22
g ( ) U ( ); g ( ) U ( ); g ( ) ( )
c c U ( ) e ( ) c c c U ( )
e ( ) c g ( ) 2 c c c U ( ) e
t t s s
t s
t t
s ? n s s ? n s s ? n[ sÇ ×/ s - [ s - n - / sÉ Ú
Ç ×- [ s -n s - n - / s -É Ú* + * +* +
* +* + * +* +
(i) (i)
33
(i) (i) (i) (i) (i)
33 31 33 31
(i) ''(i) (i) ''(i) (i) (i) (i) (i) '(i)
22 15 11 22 11 12
(i) (i) '(i) (i) (i) (i)
15 31 11 12
(i) ''(i)
15
g ( )
2e e e e ( ) 0
c U ( ) e ( ) c c 2c c U ( )
e e 1 ( ) 2 c c g ( ) 0
2e U ( )
s t
s
t
n sÇ ×- n - - - [ s ?É Ú
Ç ×s - [ s - n - - / sÉ Ú- - n - [ s - n - / s ?
s * +* + * +
* + * +
(i) ''(i) (i) (i) (i) '(i) (i) (i)
11 31 15 31 33
(i) (i) (i) (i) (i)
33 33 31
1 2 3 4
( ) e 2e e U ( ) g ( )
2 1 ( ) e 1 e U ( ) 0
, pour i 1 et , pour i 2
s
t
ÊÍÍÍÍÍÍÍÍËÍÍÍ Ç ×Í / g [ s - - n - s / g n sÉ ÚÍÍ Ç ×/ g n - [ s - n - - s ?É ÚÍÍsŒ s s ? sŒ s s ?Ì
(9)
322 J.M. Nianga and N. Recho / Procedia Materials Science 3 ( 2014 ) 318 – 324
Note that * + * +} ’ss s )()( ,ii
p UU and * +s{ )(i, respectively represent the components of the displacement and that of
electric field, in the sub-field * +2,1)( ?Y ii
, 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
2 2
2 2
2 2 2 2
(1) (2)
(1) (2)
(1) (2)
(1) (2) (1) (2)
(1) (2) (1) (2)
U U;
U U
D D;
D D
t ts?s s?ss ss?s s?s
ss ss t tts ts s ss?s s?s s?s s?s
ÊÃ Ô Ã ÔÍ ? [ ? [Ä Õ Ä ÕÄ Õ Ä ÕÍÅ Ö Å ÖÍËÃ Ô Ã Ô Ã Ô Ã Ôu uÍ ? ?Ä Õ Ä Õ Ä Õ Ä ÕÍÄ Õ Ä Õ Ä Õ Ä Õu uÅ Ö Å Ö Å Ö Å ÖÍÌ
(10)
Comparing equations (2) - (3) and (10), it follows, for 2ss ?
* +* + * +* +* + * +
* +(1) (1) (1) (1) '(1) (i) (1) (1) '(1)
12 22 22 31 15
(2) (2) (2) (2) '(2) (2) (2) (2) '(2)
12 22 22 31 15
(1) (1) '(1) (1) (1) '(1)
11 12 15
c 1 c U ( ) c U ( ) e 1 ( ) e ( )
c 1 c U ( ) c U ( ) e 1 ( ) e ( )
c c U ( ) U ( ) e (
t s
t s
t s
n - - s - s - n - [ s - [ s? n - - s - s - n - [ s - [ s
Ç ×/ s - n s - [ sÉ Ú * +(2) (2) '(2) (2) (2) '(2)
11 12 15
(1) '(1) (1) (1) (1) '(1)
15 15 11
(2) '(2) (2) (2) (2) '(2)
15 15 11
)
c c U ( ) U ( ) e ( )
2e U ( ) 2 e U ( ) ( )
2e U ( ) 2 e U ( ) ( )
t s
t st s
ÊÍÍÍÍÍËÍ Ç ×? / s - n s - [ sÉ ÚÍÍ s - n s / g [ sÍ ? s - n s / g [ sÍÌ
(11)
The notch lips being assumed free of stress and free of electric displacement, the following boundary conditions
are respectively needed for 1ss ? and 3ss ?
* +* + * +* +
(1) (1) (1) (1) '(1) (i) (1) (1) '(1)
12 22 22 31 15
(1) (1) '(1) (1) (1) '(1)
11 12 15
(1) '(1) (1) (1) (1) '(1)
15 15 11
c 1 c U ( ) c U ( ) e 1 ( ) e ( ) 0
c c U ( ) U ( ) e ( ) 0
2e U ( ) 2 e U ( ) ( ) 0
t s
t s
t s
Ê n - - s - s - n - [ s - [ s ?ÍÍ Ç ×/ s - n s - [ s ?Ë É ÚÍ s - n s / g [ s ?ÍÌ 1
( )s ? s (12)
* +* + * +* +
(2) (2) (2) (2) '(2) (2) (2) (2) '(2)
12 22 22 31 15
(2) (2) '(2) (2) (2) '(2)
11 12 15
(2) '(2) (2) (2) (2) '(2)
15 15 11
c 1 c U ( ) c U ( ) e 1 ( ) e ( ) 0
c c U ( ) U ( ) e ( ) 0
2e U ( ) 2 e U ( ) ( ) 0
t s
t s
t s
Ê n - - s - s - n - [ s - [ s ?ÍÍ Ç ×/ s - n s - [ s ?Ë É ÚÍ s - n s / g [ s ?ÍÌ3( )s ? s (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).
323 J.M. Nianga and N. Recho / Procedia Materials Science 3 ( 2014 ) 318 – 324
2. Resolution Procedure: Matrix method of interpolation for solving eigenvalue problems for ordinary
differential equations
To solve Equations (9), we propose to reformulate them, for * +21,sssΠ, in the following generic and simplified
form
] _* + * + * + * +
* + * + * + * +
k km mr r(j) (j)
ikj k ikj k 1 2
k 1 j 0 k 1 j 0
i i
i
g ( )y ( ) q ( )y ( ) 0 (i 1,...,6; , )
j 0,....,m; m 2 for k, i 1, 1 , 2, 2 , 3, 3
j 0, m 0 for k, i 4, 4 , 5, 5 , 6, 6
? ? ? ?Ê s s /n s s ? ? sŒ s sÍÍÍËÍ ? ? ?ÍÍ ? ? ?Ì
ÂÂ ÂÂ ; r = 6 (14)
Where the real numbers gijk and qijk depend on the angular variable θ and where the variables* +* +2,1;6...1 ?? mky mk are defined as follows
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 2 r
0 01
2 1
TT (m ) (m ) (m )
0 10 20 r0 1 2 rY Y ......Y ; Y Y ......Y
/Ê Ê Û Ê Û?nË Ü Ë ÜÍÍ Ì Ý Ì ÝËÍ Ç ×? ?Í É ÚÌ
Y YC C
Y Y
Y Y
(17)
The matrices C1 and C2 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* +* +rkY kmk . .1:
1/, can be then calculated with the following equation
324 J.M. Nianga and N. Recho / Procedia Materials Science 3 ( 2014 ) 318 – 324
* +k k
k
m j (m )(j)
k kj k0 k k
m 1 j
kj
Y k:1...r; j:1....m 1
0....0... ..... ......
/
/ /Ê ? - /ÍË Ç ×?Í É ÚÌY P D Y
P σ Dσ D σ (18)
3. Future extensions
The procedure developed here can be applied to any 2D-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 University
Press. 2. J. Li , N. Recho (2002). Méthodes asymptotiques en mécanique de la rupture (in French). Editions Hermes Sciences, Paris.
3. H. A. Sosa and Y. E. Pak coordinated, 1990. Three dimensional eigenfunctions analysis of a crack in apiezoelectric material. Int. J. Solids.
Struct. 26, 1, 15.
4. 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. 33-50
5. Z. Suo., C.-M. Kuo., D. M. Barnett., J. R. Willis., 1992. Fracture Mechanics forpiezoelectric ceramics. J. Mech. Phys. Solids, vol. 40, no.
4, pp. 739-765.
6. Z. Niu., D. Ge., C. Cheng., J. Ye, N. Recho, 2009. Evaluation of the stress singularities of plane V-notches in bonded dissimilar materials, Applied Mathematical Modelling, vol. 33, no. 3, pp. 1776–1792
7. Z. Niu., C. Cheng., J. Ye, and N. Recho, 2009. “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. 2999–3008.
