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The interface crack problem for a piezoelectric semi-infinite strip under concentrated electromechanical loading
Engineering Fracture Mechanics 71 (2004) 1853–1871 www.elsevier.com/locate/engfracmech
The interface crack problem for a piezoelectric semi-infinite strip under concentrated electromechanical loading V. Govorukha a
a,*
, M. Kamlah b, D. Munz
b
Department of Computational Mathematics, Dnepropetrovsk National University, 49050 Dnepropetrovsk, Ukraine b Forschungszentrum Karlsruhe, Institut for Materials Research II, Postfach 3640, 76021 Karlsruhe, Germany Received 19 May 2003; received in revised form 15 October 2003; accepted 2 December 2003
Abstract A plane strain problem for an interface crack along the fixed edge of a piezoelectric semi-infinite strip under concentrated electromechanical loading is examined. The model of an interface crack with an artificial contact zone near its tips is used. By using Fourier transforms, boundary integral relations are derived. By means of these relations, the system of singular integral equations for several electric boundary conditions are formulated and solved numerically. The stress intensity factors at the crack tip and the energy release rate are calculated. Influence of the permeability of the crack on electric and mechanical fields near the crack tip is considered. 2004 Elsevier Ltd. All rights reserved. Keywords: Piezoelectric material; Interface crack; Contact zone; Permittivity of crack medium
1. Introduction Piezoelectric materials exhibit an interesting phenomenon in that they produce an electric field when deformed and undergo deformation when subjected to an electric field. An overview over microscopic and phenomenological approaches related to electromechanical coupling phenomena is given in [1]. Due to this intrinsic electromechanical coupling behaviour, piezoelectric materials have been widely used in technology as sensors and actuators. On the other hand piezoelectric materials are very brittle and susceptible to fracture. Therefore, in order to predict the performance and the integrity of piezoelectric devices, it is important that the behaviour of various defects are analyzed and studied under electrical and mechanical fields. A wealth of theoretical results, in the field of piezoelectric fracture, has been presented in [2–11]. They have solved basic problems for cracks in homogeneous piezoelectric materials and introduced some fracture criteria for piezoelectric ceramics. However, the problem of an interface crack in a piezoelectric bimaterial, *
Corresponding author. E-mail address: [email protected] (V. Govorukha).
0013-7944/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2003.12.005
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Nomenclature rij stress components sij strain components Ei electric field components Di electric displacement components u, w components of the displacement vector u electric potential fi body force density q body charge density cij elastic constants eij dielectric constants eij piezoelectric constants ðn; gÞ coordinates of the concentrated electromechanical loading Fx , Fz components of the concentrated force q0 magnitude of the concentrated electric charge 2d thickness of the interface crack ea permittivity of the medium inside the interface crack er relative permittivity of the medium 2h width of the piezoelectric semi-infinite strip 2b length of the interface crack a < jxj < b frictionless contact zones at the tip of the crack k ¼ ðb aÞ=2b relative length of the contact zone k0 real contact zone length Ki stress intensity factors G energy release rate K20 ¼ K2 ðk0 Þ stress intensity factor K2 corresponding to the real contact zone length G0 ¼ Gðk0 Þ energy release rate corresponding to the real contact zone length All other symbols are mathematical constants or represent abbreviations for mathematical expressions and do not denote fundamental physical quantities.
in spite of its importance, has not obtained due attention in the literature because of its complexity. The first attempt to analyze piezoelectric interface crack problems from a fracture mechanics point of view appears to be that of Parton [12], who analyzed a finite crack at the interface of a piezoelectric material and a conductor subjected to a far-field uniform tension. Later, Suo et al. [13] examined the problem of an interface crack between dissimilar anisotropic piezoelectric media and the dependence of singularities at the tips of an interface crack with respect to different electrical conditions. A new type of singularity has been found in these papers. Beom and Atluri [14] investigated a complete form of stress and electric displacement field in the vicinity of an interface crack in dissimilar anisotropic piezoelectric materials, and defined a new set of stress and electric displacement intensity factors. The generalized two-dimensional problem of collinear interfacial cracks between two dissimilar piezoelectric media subjected to piecewise uniform loads at infinity has been studied by Gao and Wang [15]. The singular integral equations approach to the interface crack problem for piezoelectric materials has been studied by Govorukha et al. [16]. Weight function analysis for interface cracks in dissimilar anisotropic piezoelectric materials has been proposed and investigated by Ma and Chen [17].
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It should be noted that in most of the above-mentioned investigations were performed in the frame of the classical interface crack model [18], initially assuming that the interface crack is completely opened and leading to an oscillating singularity at the crack tips. But along with the classical interface crack model, the contact zone model [19] was developed for a crack between two non-piezoelectric materials. This model was applied to interface cracks in thermo-piezoelectric materials by Qin and Mai [20]. Besides, the usefulness of a new interface crack model based on an artificial contact zone was demonstrated by Herrmann and Loboda [21] and in particular cases, classical model and contact zone model, were found. On the basis of this model a detailed analytical investigation of interface crack between two piezoelectric semiinfinite planes with respect to different electrical conditions has been performed in recent investigations [22,23]. In the theoretical analyses of the crack problem in piezoelectric materials, the researchers assume the crack surfaces to be stress-free, but they have different opinions about the electrical boundary condition at the crack surfaces. As the dielectric constant of air or the medium between the crack faces is very small compared to that of the piezoelectric material Deeg [2], Pak [5] have assumed crack surfaces to be free of surface charge. On the other hand Parton [12] has assumed that although the magnitude of the normal electrical displacement component at the crack face is very small, the electric displacement is continuous across the crack faces. Using the analogy of a capacitor filled by a medium, Hao and Shen [24] used a boundary condition in which the electrical permeability of air in the crack gap is considered. Detailed discussion of this topic is presented in the following sections. In most of the above studies the mechanical and electrical loading is chosen to be acting upon the crack faces or at infinity. But in some practical cases, a mechanical or a electrical load can be applied at the internal points of the domain. To the best of our knowledge, correspondent results for piezoelectric interface crack problems have not been obtained yet. In this paper the interface problem for a crack along the fixed end of an piezoelectric semi-infinite strip is investigated with taking into account the permittivity of the crack medium. The model of a crack with an artificial contact zone near the tips is utilized. The problem is reduced to a system of singular integral equations corresponding to different electrical conditions. The numerical solution of these systems of equations gives the possibility to find main fracture characteristics for all the considered cases.
2. Basic equations The constitutive equations for the piezoelectric materials can be written as [3] rij ¼ cijkl skl ekij Ek ;
ð1Þ
Di ¼ eikl skl þ eik Ek ;
ð2Þ
where rij , skl , Ek and Di , are stress, strain, electric field and electric displacement, respectively. cijkl , eik and eikl are the elastic constants, the dielectric constants and the piezoelectric constants tensor of the material, respectively. The strain tensor is related to the displacement vector fui ; i ¼ 1; 2; 3g by the equation sij ¼ ðui;j þ uj;i Þ=2;
ð3Þ
where a comma indicates the partial derivative. The electric field strength is related to the electric potential u by the equation Ei ¼
ou : oxi
ð4Þ
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The governing field equations are given by rij;j ¼ fi ;
ð5Þ
Di;i ¼ q;
ð6Þ
where fi and q are the body force and body charge, respectively. Poled piezoelectric materials like barium titanate (BaTiO3 ) and lead zirconate titanate (PZT) are transversely isotropic materials with z in the poling direction. For the latter, a plane strain response is considered and a Cartesian coordinate system ðx; y; zÞ is positioned in such a manner that the plane deformation is orthogonal to the y-axis. Eqs. (1)–(6) are written for this coordinate system. Then Eqs. (3) and (4) are substituted into Eqs. (1) and (2) and the result is put into Eqs. (5) and (6). The governing differential equations now become o2 u o2 u o2 w o2 u þ ðe ¼ Fx dðx nÞdðz gÞ; þ c þ ðc þ c Þ þ e Þ 44 13 44 31 15 ox2 oz2 ox oz ox oz o2 u o2 w o2 w o2 u o2 u þ c44 2 þ c33 2 þ e33 2 þ e15 2 ¼ Fz dðx nÞdðz gÞ; ðc13 þ c44 Þ ox oz ox oz oz ox o2 u o2 w o2 w o2 u o2 u þ e15 2 þ e33 2 e11 2 e33 2 ¼ q0 dðx nÞdðz gÞ; ðc13 þ c44 Þ ox oz ox oz ox oz
c11
ð7Þ
where fFx ; Fz g are components of the concentrated force and q0 is the magnitude of the concentrated electric charge, which are applied at point ðn; gÞ. dðxÞ is the Dirac-delta function. In this system the displacement components uðx; zÞ, wðx; zÞ, and the electric potential uðx; zÞ are the unknown functions.
3. The electrical boundary conditions on the faces of the crack It has already been noted that several electric boundary conditions were proposed in literature. They differ in the shielding strength of the electric displacement at the crack. A useful approach to the formulation of such conditions was suggested in [3]. The boundary conditions were formulated under the assumption that the surfaces of contacting bodies are separated by a thin intermediate layer describable in terms of certain reduced physical parameters. If the parameters are considered to be constant, the conditions for the potential u and the electric displacement normal component Dz , on the physical surface of the transition layer read as ex
o2 þ ðu þ u Þ 2ðDþ z Dz Þ ¼ 0; ox2
ð8Þ
ex
o2 þ þ ðu u Þ 6ðDþ z Dz Þ 12ez ðu u Þ ¼ 0: ox2
ð9Þ
Superscripts ‘‘+’’ and ‘‘)’’ denote the upper and lower crack surfaces, respectively. It is seen that the electrical boundary conditions on the surface of the layer are characterized by the two parameters, ex ¼ ea 2d;
ez ¼ ea =ð2dÞ;
ð10Þ
where 2d is the thickness and ea is the permittivity of the layer, respectively. Eqs. (8) and (9) may be used for deriving the electrical boundary conditions at the faces of a crack in a piezoelectric medium. Some special cases are as follows:
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(i) If ex ! 0 and ez ! 1, Eqs. (8) and (9) lead to the permeable boundary condition Dþ z ¼ Dz ;
uþ ¼ u ;
ð11Þ
which does not shield the electric displacement at all. (ii) When ex ¼ ez ¼ 0, we obtain the impermeable boundary condition Dþ z ¼ Dz ¼ 0;
ð12Þ
which is supposed to shield the electric displacement completely. (iii) Letting ex ¼ 0 and ez 6¼ 0 yields the boundary condition, in which the shielding of the crack depends on the permittivity ea of the medium inside the crack, Dþ z ¼ Dz ¼ Dz ;
uþ u ¼
1 Dz : ez
ð13Þ
This boundary condition is called limited permeable. It is worth noting that the boundary condition (13) is related to the boundary condition [24]. In the formulation used here, the thickness d is a constant. However, this restriction appeared to be necessary, because the solution of the problems considered in the following sections, seems to be possible only on the basis of the form (13) of the limited permeable boundary condition with constant thickness. The thickness d will be assigned a physically reasonable value in the context of the solution of the problem in Section 5.3. In this paper the medium is defined by its relative permittivity er , which is the ratio between the permittivity ea and the permittivity of vacuum, e0 ¼ 8:85 1012 A s/V m: ea ¼ er e0 :
ð14Þ
For air and vacuum, er ¼ 1. There is no physical medium with er < 1. The permeable and impermeable boundary conditions may be considered as the extreme cases of the limited permeable boundary condition.
4. Formulation of boundary integral relations for a semi-infinite piezoelectric strip Consider a plane strain state in the plane ðx; zÞ for a piezoelectric semi-infinite strip shown in Fig. 1. Let in the points Aðn; gÞ and A0 ðn; gÞ concentrated forces and electric charges be symmetrically applied. The boundary conditions on the sides x ¼ h of the semi-infinite strip are given in the form uðh; zÞ ¼ 0;
ow ðh; zÞ ¼ 0; ox
ou ðh; zÞ ¼ 0: ox
ð15Þ
The electromechanical behaviour of the considered material under concentrated loads is fully described by the governing equations (7). In order to solve the differential equations (7) we introduce the finite Fourier transforms uðp; zÞ ¼
Z
h
uðx; zÞ sinðxpxÞ dx; 0
ðp; zÞ ¼ w
Z
h
wðx; zÞ cosðxpxÞ dx;
ðp; zÞ ¼ u
0
Z
h
uðx; zÞ cosðxpxÞ dx 0
ð16Þ
are governed by the ordinary differential equations and u in which case u, w
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Fig. 1. The piezoelectric semi-infinite strip of width 2h under concentrated electromechanical loading. The ranges ða; bÞ represents the frictionless contact zone. The location of the concentrated force fFx ; Fz g and electric charge q0 is ðn; gÞ.
d2 d w d u u xpðe31 þ e15 Þ ¼ Fx sinðxpnÞdðz gÞ; xpðc13 þ c44 Þ dz dz dz2 d u d2 w d2 u þ e33 2 ¼ Fz cosðvpnÞdðz gÞ; þ c33 2 x2 p2 e15 u xpðc13 þ c44 Þ x2 p2 c44 w dz dz dz 2 2 d u d d w u e33 þ e33 2 þ x2 p2 e11 u ¼ q0 cosðxpnÞdðz gÞ; xpðe15 þ e31 Þ x2 p2 e15 w dz dz dz x2 p2 c11 u þ c44
ð17Þ
where x ¼ p=h (we have taken into account that due to symmetry uð0; zÞ ¼ 0, owð0; zÞ=ox ¼ 0, ouð0; zÞ=ox ¼ 0). Applying further Fourier transforms with respect to the z-coordinate, Z 1 Z 1 Z 1 ^ ðp; tÞ ¼ ðp; zÞ sinðtzÞ dz; ^ ðp; tÞ ¼ ðp; zÞ sinðtzÞ dz; ^ uðp; zÞ cosðtzÞ dz; w w u u uðp; tÞ ¼ 0
0
0
^: ^ and u to Eq. (17), we arrive at the following algebraic system with respect to ^u, w ðx2 p2 c11 þ c44 t2 Þ^ u þ xptðc13 þ c44 Þ^ w þ xptðe31 þ e15 Þ^ u ¼ q1 þ xpc13 Q2 þ xpe31 Q3 þ Fx sinðxpnÞ cosðtgÞ; u þ ðx2 p2 c44 þ c33 t2 Þ^ w þ ðx2 p2 e15 þ t2 e33 Þ^ u ¼ tc33 Q2 þ te33 Q3 þ Fz cosðxpnÞ sinðtgÞ; xptðc13 þ c44 Þ^ xptðe15 þ e31 Þ^ u þ ðx2 p2 e15 þ e33 t2 Þ^ w ðx2 p2 e11 þ t2 e33 Þ^ u ¼ te33 Q2 te33 Q3 q0 cosðxpnÞ sinðtgÞ:
ð18Þ
Here, Q2 ðpÞ ¼
Z
h
Q2 ðxÞ cosðxpxÞ dx;
Q3 ðpÞ ¼
0
Z
h
Q3 ðxÞ cosðxpxÞ dx; 0
and q1 ðxÞ ¼ rxz ðx; 0Þ;
Q2 ðxÞ ¼ wðx; 0Þ;
Q3 ðxÞ ¼ uðx; 0Þ:
q1 ðpÞ ¼
Z
h
q1 ðxÞ sinðxpxÞ dx 0
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The form of the solution of this system depends on the roots of the characteristic equation At6 þ Bt4 p2 x2 þ Ct2 p4 x4 þ Dp6 x6 ¼ 0;
ð19Þ
where A ¼ c44 ðc33 e33 þ e233 Þ; B ¼ 2e33 ðc13 þ c44 Þðe31 þ e15 Þ c11 c33 e33 c244 e33 c44 c33 e11 c33 ðe15 þ e31 Þ2 2
2e33 e15 c44 e233 c11 þ e33 ðc13 þ c44 Þ ; C ¼ 2e15 ðc13 þ c44 Þðe31 þ e15 Þ c11 c44 e33 c11 c33 e11 c244 e11 c44 ðe15 þ e31 Þ
2
e215 c44 2e15 e33 c11 þ e11 ðc13 þ c44 Þ2 ; D ¼ c11 ðc44 e11 þ e215 Þ: Analysis of Eq. (19) shows that for realistic values of the material constants of the considered class of materials, this equation has two real roots k1 and two pairs of complex conjugate roots ðd ihÞ [3]. Using the inverse Fourier transforms for the solution of the system (18) leads to the following expressions: 3 Z h ou 1 X o~u ðx; zÞ ¼ f2c1i R1 ðx; z; tÞ þ c2i R2 ðx; z; tÞ þ c3i R3 ðx; z; tÞgqi ðtÞ dt þ ðx; zÞ; ox 2h i¼1 0 ox 3 Z h ow 1 X o~ w ðx; zÞ ¼ ðx; zÞ; f2q1i R1 ðx; z; tÞ þ q2i R2 ðx; z; tÞ þ q3i R3 ðx; z; tÞgqi ðtÞ dt þ oz 2h i¼1 0 oz
ð20Þ
3 Z h ou 1 X o~ u ðx; zÞ ¼ ðx; zÞ; f2v1i R1 ðx; z; tÞ þ v2i R2 ðx; z; tÞ þ c3i R3 ðx; z; tÞgqi ðtÞ dt þ oz 2h i¼1 0 oz
where q2 ðtÞ ¼ Q02 ðtÞ; R1 ðx; z; tÞ ¼
R2 ðx; z; tÞ ¼
q3 ðtÞ ¼ Q03 ðtÞ;
sin½xðt þ xÞ sin½xðt xÞ þ ; cosh½xa1 z cos½xðt þ xÞ cosh½xa1 z cos½xðt xÞ 2 X n¼1
R3 ðx; z; tÞ ¼
2 X
n n sin½xðt þ x þ ð1Þ a5 zÞ sin½xðt x þ ð1Þ a5 zÞ þ ; n n cosh½xa4 z cos½xðt þ x þ ð1Þ a5 zÞ cosh½xa4 z cos½xðt x þ ð1Þ a5 zÞ
ð1Þ
nþ1
n¼1
a21 ¼ jk1 j;
a22 ¼ jdj;
sinh½xa4 z sinh½xa4 z þ ; n n cosh½xa4 z cos½xðt þ x þ ð1Þ a5 zÞ cosh½xa4 z cos½xðt x þ ð1Þ a5 zÞ
a23 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 2 þ h2 ;
a24 ¼ ða23 þ a22 Þ=2;
a25 ¼ ða23 a22 Þ=2:
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The following values are defined by the external load: 2 o~ u 1 X ðx; zÞ ¼ ½f~c Li ðx; z; n; gÞ þ ~c12 Xi ðx; z; n; gÞ þ ~c13 Yi ðx; z; n; gÞgFx ox 4h i¼1 11
þ f~c21 Ni ðx; z; n; gÞ þ ~c22 Xi ðx; z; n; gÞ þ ~c23 Yi ðx; z; n; gÞgFz þ f~c31 Ni ðx; z; n; gÞ þ ~c32 Xi ðx; z; n; gÞ þ ~c33 Yi ðx; z; n; gÞgq0 ; 2 o~ w 1 X ~12 Xi ðx; z; n; gÞ þ q ~13 Yi ðx; z; n; gÞgFx ðx; zÞ ¼ ½f~ q11 Li ðx; z; n; gÞ þ q oz 4h i¼1
~22 Xi ðx; z; n; gÞ þ q ~23 Yi ðx; z; n; gÞgFz þ f~ q21 Ni ðx; z; n; gÞ þ q
ð21Þ
~32 Xi ðx; z; n; gÞ þ q ~33 Yi ðx; z; n; gÞgq0 ; þ f~ q31 Ni ðx; z; n; gÞ þ q 2 o~ u 1 X ðx; zÞ ¼ ½f~ v Li ðx; z; n; gÞ þ ~ v12 Xi ðx; z; n; gÞ þ ~v13 Yi ðx; z; n; gÞgFx oz 4h i¼1 11
þ f~ v21 Ni ðx; z; n; gÞ þ ~ v22 Xi ðx; z; n; gÞ þ ~v23 Yi ðx; z; n; gÞgFz þ f~ v31 Ni ðx; z; n; gÞ þ ~ v32 Xi ðx; z; n; gÞ þ ~v33 Yi ðx; z; n; gÞgq0 ; where Li ðx; z; n; gÞ ¼
sin½xðn þ xÞ sin½xðn xÞ þ ; i cosh½xa1 ðz þ ð1Þ gÞ cos½xðn þ xÞ cosh½xa1 ðz þ ð1Þi gÞ cos½xðn xÞ
sinh½xa1 ðz þ ð1Þi gÞ sinh½xa1 ðz þ ð1Þi gÞ þ ; i i cosh½xa1 ðz þ ð1Þ gÞ cos½xðn þ xÞ cosh½xa1 ðz þ ð1Þ gÞ cos½xðn xÞ ( 2 X sin½xðn þ x þ ð1Þn a5 ðz þ ð1Þi gÞÞ Xi ðx; z; n; lÞ ¼ i n i cosh½xa4 ðz þ ð1Þ gÞ cos½xðn þ x þ ð1Þ a5 ðz þ ð1Þ gÞÞ n¼1 ) n i sin½xðn x þ ð1Þ a5 ðz þ ð1Þ gÞÞ þ ; cosh½xa4 ðz þ ð1Þi gÞ cos½xðn x þ ð1Þn a5 ðz þ ð1Þi gÞÞ Ni ðx; z; n; gÞ ¼
Yi ðx; z; n; gÞ ¼
2 X
(
i
sinh½xa4 ðz þ ð1Þ gÞ i cosh½xa ðz þ ð1Þ gÞ cos½xðn þ x þ ð1Þn a5 ðz þ ð1Þi gÞÞ 4 n¼1 ) i sinh½xa4 ðz þ ð1Þ gÞ þ : i n i cosh½xa4 ðz þ ð1Þ gÞ cos½xðn x þ ð1Þ a5 ðz þ ð1Þ gÞÞ
~ij , ~ vij (i; j ¼ 1; 2; 3) are defined by the electric and elastic parameters of the The real constants cij , qij , vij , ~cij , q material. Their actual expressions are quite lengthy and are omitted because they do not introduce any relevant information in the final results. Expressions for the entries in o~u=ox, o~ w=oz and o~ u=oz shown above are valid for z P g. For z 6 g, these expressions are expressible by interchanging z with g. Extracting the singular components of the integrals in (20), we obtain the following formulas for strain, stress and electric displacement on edge of the semi-infinite strip: Z h 3 1X 1 u0 ðx; 0Þ ¼ b1i ð22Þ þ Mðx; tÞ qi ðtÞ dt þ ~u0 ðx; 0Þ; p i¼1 h t x
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rzz ðx; 0Þ ¼
Z h 3 1X 1 ~zz ðx; 0Þ; b2i þ Mðx; tÞ qi ðtÞ dt þ r p i¼1 h t x
ð23Þ
Dz ðx; 0Þ ¼
Z h 3 1X 1 ~ z ðx; 0Þ; b3i þ Mðx; tÞ qi ðtÞ dt þ D p i¼1 h t x
ð24Þ
where u0 ðx; 0Þ ¼
ou ðx; 0Þ; ox
b1i ¼ c1i þ c2i ; b2i ¼ c13 ðc1i þ c2i Þ þ c33 ðq1i þ q2i Þ þ e33 ðv1i þ v2i Þ; b3i ¼ e31 ðc1i þ c2i Þ þ e33 ðq1i þ q2i Þ þ e33 ðv1i þ v2i Þ; ~zz ðx; 0Þ ¼ c13 r
o~ u o~ w o~ u ðx; 0Þ þ c33 ðx; 0Þ þ e33 ðx; 0Þ; ox oz oz
u o~ w o~ u ~ z ðx; 0Þ ¼ e13 o~ D ðx; 0Þ þ e33 ðx; 0Þ e33 ðx; 0Þ; ox oz oz x sin½xðt þ xÞ sin½xðt xÞ 2 Mðx; tÞ ¼ þ ; 2 1 cos½xðt þ xÞ 1 cos½xðt xÞ xðt xÞ
i ¼ 1; 2; 3:
In the last expressions the function Mðx; tÞ 2 H , where H is the class of H€ older functions [25]. The boundary integral relations (22)–(24) play an important role in the following analysis because by means of these relations the singular integral equations for various boundary conditions at the interface can be formulated.
5. An interface crack under concentrated electromechanical loadings 5.1. Solution of the permeable crack problem Suppose that a semi-infinite piezoelectric strip is perfectly bounded at the edge z ¼ 0, b < jxj < h to a rigid conductor and an interface crack is situated along the interval z ¼ 0, jxj < b. In this section, we consider the permeable crack problem. The boundary conditions are the following: uðh; zÞ ¼ 0; rxz ðx; 0Þ ¼ 0;
ow ou ðh; zÞ ¼ 0; ðh; zÞ ¼ 0; ox ox rzz ðx; 0Þ ¼ 0; jxj < b;
uðx; 0Þ ¼ 0;
wðx; 0Þ ¼ 0;
uðx; 0Þ ¼ 0;
jxj < h:
ð25Þ
b < jxj < h;
According to (25) one can set in Eqs. (22)–(24) q1 ðxÞ ¼ 0 for jxj < b, q2 ðxÞ ¼ 0 for b < jxj < h and q3 ðxÞ ¼ 0. Also making use of the symmetry properties of these unknown functions and satisfying further the equations u0 ðx; 0Þ ¼ 0 and rzz ðx; 0Þ ¼ 0, we obtain the following system of singular integral equations (SIE):
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Z bi1 b
h
Z b 1 1 1 þ M ðx; tÞ q1 ðtÞ dt þ bi2 þ þ Mðx; tÞ q2 ðtÞ dt ¼ ri ; tx tþx b t x
ð26Þ
where x 2 ðb; hÞ for i ¼ 1, and x 2 ðb; bÞ for i ¼ 2, M ðx; tÞ ¼ Mðx; tÞ Mðx; tÞ, r1 ¼ p~u0 ðx; 0Þ, r2 ¼ p~ rzz ðx; 0Þ. In this case we obtain singularities [18], which imply oscillations in the stress field near the tips of the crack, resulting in the unrealistic phenomenon of interpenetrating crack faces, and leading to difficulties in the numerical analysis for the determination of the fracture parameters. For this reason, similarly to [19] we introduce frictionless contact zones a < jxj < b to avoid the oscillating singularity, where the position of the point a is arbitrary for the time being. For such an arbitrary position of the point a, we have an artificial contact zone model, which is not physically justified, but from this model the specific value of a for the real contact zone length can be found. The boundary conditions at z ¼ 0 for this model are the following: rxz ðx; 0Þ ¼ 0; rxz ðx; 0Þ ¼ 0;
rzz ðx; 0Þ ¼ 0; wðx; 0Þ ¼ 0;
uðx; 0Þ ¼ 0; wðx; 0Þ ¼ 0; uðx; 0Þ ¼ 0; jxj < h:
jxj < a; a < jxj < b;
ð27Þ
b < jxj < h;
In this case q1 ðxÞ ¼ 0 for jxj < b, q2 ðxÞ ¼ 0 for a < jxj < h and q3 ðxÞ ¼ 0, and we arrive at the very same system of SIE (26), in which now x 2 ðb; hÞ for i ¼ 1, and x 2 ða; aÞ for i ¼ 2. Additional conditions must be satisfied to ensure that the displacements are single-valued: Z a Z b ouðx; 0Þ owðx; 0Þ dx ¼ 0; dx ¼ 0: ox ox b a When making use of relation (22), these conditions can be written in the form Z h Z a 1 cos½xðt þ bÞ 1 cos½xðt þ bÞ q2 ðtÞ dt q1 ðtÞ dt þ b12 2b11 ln ln 1 cos½xðt bÞ 1 cos½xðt bÞ b a Z a q2 ðtÞ dt ¼ 0: ¼ 2p~ uðb; 0Þ
ð28Þ
a
Due to the absence of the oscillating singularity in this case, the unknown functions qi ðtÞ (i ¼ 1; 2) can be represented in the form q1 ðtÞ q1 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðh tÞðt bÞ
q2 ðtÞ ffi; q2 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi a2 t 2
qi ðtÞ 2 H :
Next we introduce the stress intensity factors (SIF) for the right crack tip as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K1 ¼ lim 2ðx aÞrzz ðx; 0Þ; 2ðx bÞrxz ðx; 0Þ: K2 ¼ lim x!aþ0
x!bþ0
Due to (23), (29) and the results [25] for Cauchy type integral behaviour, we can write rffiffiffiffiffiffiffiffiffiffiffi b22 q2 ðaÞ 2 K1 ¼ pffiffiffi ; K2 ¼ q ðbÞ: hb 1 a
ð29Þ
ð30Þ
ð31Þ
The energy release rate (ERR) G for the right crack tip can be presented by the virtual work integral [3] Z aþDl Z bþDl 1 rzz ðx; 0Þwðx þ Dl; 0Þ dx þ rxz ðx; 0Þuðx þ Dl; 0Þ dx : ð32Þ G ¼ lim Dl!0 2 Dl a b
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The integrals in the last formula can be computed from the asymptotic behaviour of displacements and stresses near the singular points a and b. Using the asymptotic behaviour of q2 ðxÞ for x ! a 0 we find from the second of relations (16) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ða xÞ q2 ðaÞ: wðx; 0Þjx!a0 ¼ a Employing Cauchy type properties in Eq. (23) gives q2 ðaÞ rzz ðx; 0Þjx!aþ0 ¼ b22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2aðx aÞ In a similar way it can be shown that rffiffiffiffiffiffiffiffiffiffiffi bx uðx; 0Þjx!b0 ¼ 2b11 q ðbÞ; hb 1 q1 ðbÞ ffi: rxz ðx; 0Þjx!bþ0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðh bÞðx bÞ Substituting the last formulas into (32) and taking into account that Z Dl rffiffiffiffiffiffiffiffiffiffiffiffiffi Dl x p Dl dx ¼ ; x 2 0 we obtain the following formula: p b22 2 2b11 2 G¼ q2 ðaÞ q1 ðbÞ : 4 a hb
ð33Þ
5.2. Solution of the impermeable crack problem Consider now the impermeable crack problem. The boundary conditions for this case are the following: uðh; zÞ ¼ 0; rxz ðx; 0Þ ¼ 0; uðx; 0Þ ¼ 0; Dz ðx; 0Þ ¼ 0;
ow ðh; zÞ ¼ 0; ox rzz ðx; 0Þ ¼ 0; wðx; 0Þ ¼ 0; jxj < b;
ou ðh; zÞ ¼ 0; ox jxj < b;
ð34Þ
b < jxj < h; uðx; 0Þ ¼ 0;
b < jxj < h:
In this case one can set in Eqs. (22)–(24) q1 ðxÞ ¼ 0 (jxj < b), q2 ðxÞ ¼ 0 (b < jxj < h) and q3 ðxÞ ¼ 0 (b < jxj < h). Further, we arrive at the following system of three SIEs: Z h Z b 3 X 1 1 1 þ M ðx; tÞ q1 ðtÞ dt þ bi1 bik ð35Þ þ þ Mðx; tÞ qk ðtÞ dt ¼ ri : tx tþx b b t x k¼2 e z ðx; 0Þ, x 2 ðb; hÞ for i ¼ 1, and x 2 ðb; bÞ for i ¼ 2; 3. Here, r3 ¼ p D The analysis of this system [16] showed that in the case of impermeable crack faces of a piezoelectric composite both the oscillating singularity and the real singularity of a power type appears. To avoid the oscillating singularity similar to the case of the permeable faces, we may introduce the frictionless contact zones a < jxj < b at the tips of the crack.
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The system of singular integral equations in this case has the form Z h Z a 3 X 1 1 1 þ M ðx; tÞ q1 ðtÞ dt þ bik bi1 þ þ Mðx; tÞ qk ðtÞ dt ¼ ri ; tx tþx b a t x k¼2 where x 2 ðb; hÞ for i ¼ 1, and x 2 ða; aÞ for i ¼ 2; 3. Additional conditions take the following form: Z h Z a 3 X 1 cos½xðt þ bÞ 1 cos½xðt þ bÞ q1 ðtÞ dt þ ln b ln 2b11 1k 1 cos½xðt bÞ qk ðtÞ dt ¼ 2p~uðb; 0Þ 1 cos½xðt bÞ b a k¼2 Z a qk ðtÞ dt ¼ 0 ðk ¼ 2; 3Þ:
ð36Þ
ð37Þ
a
We present the solution of the system (36) in the form q1 ðtÞ q1 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðh tÞðt bÞ
qk ðtÞ ffi qk ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi a2 t 2
ðk ¼ 2; 3Þ:
The SIFÕs K1 , K2 can be found by the same method as before and have the form rffiffiffiffiffiffiffiffiffiffiffi 1 2 K1 ¼ pffiffiffi ½b22 q2 ðaÞ þ b23 q3 ðaÞ; K2 ¼ q ðbÞ: hb 1 a
ð38Þ
ð39Þ
The ERR was found by the virtual work integral Z aþDl Z bþDl 1 rzz ðx; 0Þwðx þ Dl; 0Þ dx þ rxz ðx; 0Þuðx þ Dl; 0Þ dx G ¼ lim Dl!0 2 Dl a b Z aþDl þ Dz ðx; 0Þuðx þ Dl; 0Þ dx a
and was similarly to (32) evaluated in the form p 2ab11 2 2 2 b22 q2 ðaÞ þ b33 q3 ðaÞ G¼ q ðbÞ þ ðb23 þ b32 Þq2 ðaÞq3 ðaÞ : 4a hb 1
ð40Þ
5.3. Solution of the limited permeable crack problem The permeable and impermeable boundary conditions at the crack faces of non-conducting cracks in piezoelectric materials are usually used in literature. But both variants are too one-sided to model such cracks realistically. Instead, the electric permeability of the crack medium should be taken into account. That is why we will assume next that the crack faces are limited permeable. The boundary conditions for this crack with an artificial contact zone are the following: uðh; zÞ ¼ 0;
ow ðh; zÞ ¼ 0; ox
rxz ðx; 0Þ ¼ 0;
rzz ðx; 0Þ ¼ 0;
rxz ðx; 0Þ ¼ 0;
wðx; 0Þ ¼ 0;
uðx; 0Þ ¼ 0; Dz ðx; 0Þ ¼
wðx; 0Þ ¼ 0;
ez uðx; 0Þ;
ou ðh; zÞ ¼ 0; ox jxj < a; ð41Þ
a < jxj < b; b < jxj < h;
jxj < a;
uðx; 0Þ ¼ 0;
a < jxj < h:
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Taking into account that in this case q1 ðxÞ ¼ 0 (jxj < b), q2 ðxÞ ¼ 0 (a < jxj < h) and q3 ðxÞ ¼ 0 (a < jxj < h) and satisfying the remaining boundary conditions, we obtain the system Z h Z a 3 X 1 1 1 þ M ðx; tÞ q1 ðtÞ dt þ bik bi1 þ þ Mðx; tÞ qk ðtÞ dt ¼ ri ði ¼ 1; 2Þ; tx tþx b a t x k¼2 Z h Z Z x 3 a X 1 1 1 þ M ðx; tÞ q1 ðtÞ dt þ b3k q3 ðtÞ dt b31 þ þ Mðx; tÞ qk ðtÞ dt ¼ r3 pez tx tþx b a t x a k¼2 ð42Þ of SIEs, where x 2 ðb; hÞ for i ¼ 1, and x 2 ða; aÞ for i ¼ 2; 3. Additional conditions are taken in the form (37). The solution of the system (42) was found in the form of (38) and in order to realize the non-linear boundary conditions, an iteration scheme ne o a fez gkþ1 ¼ ð43Þ 2d k has been developed. The thickness d in the boundary condition (13) is assigned the mean crack opening: Z a 1 wðx; 0Þ dx: d¼ 2a a The iteration scheme starts with an impermeable crack assumption (iteration number 0), which gives a crack opening displacement and the right-hand side of the iteration equation (43) can be evaluated. The new field problem is solved again, leading to a new crack opening displacement, which will be once more inserted into the right-hand side of the iteration equation, and so on. The SIFs K1 and K2 have the same sense here as in (39). The ERR is defined by Z aþDl Z bþDl 1 rzz ðx; 0Þwðx þ Dl; 0Þ dx þ rxz ðx; 0Þuðx þ Dl; 0Þ dx G ¼ lim Dl!0 2 Dl a b Z aþDl Z aþDl Dz ðx; 0Þuðx þ Dl; 0Þ dx Dz ðx þ DlÞuðx þ Dl; 0Þ dx þ a
a
and can be found by the same method as earlier in the form p 2b33 2 2ab11 2 2b32 2 b22 q2 ðaÞ þ G¼ q ðaÞ q ðbÞ þ b23 þ q2 ðaÞq3 ðaÞ : 4a p 3 hb 1 p
ð44Þ
Here, uðx; 0Þ is defined as Z x uðx; 0Þ ¼ q3 ðtÞ dt: a
6. Numerical results and discussion A numerical solution of the system SIE has been obtained by the method based on the Gauss–Chebyshev quadrature rule, which was described in detail in [26]. The piezoceramic BaTiO3 is considered for the numerical analyses and its material constants are given in [27]. The results of the calculation of the SIFs K1 , K2 and ERR G for permeable crack faces are given in Table 1. The obtained results are valid in the mathematical sense for any values of the parameter
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Table 1 Dependence of the stress intensity factors K1 , K2 and the energy release rate G on the relative contact zone length k for permeable crack faces pffiffiffi pffiffiffi K2 =ðFz bÞ G=ðFz bÞ 2k K1 =ðFz bÞ 102 103 104
0.261 0.256 0.251
k¼
)0.0569 )0.0811 )0.0919
0.128 · 1011 0.129 · 1011 0.129 · 1011
ba : 2b
However, this solution is physically correct only for the particular case k ¼ k0 , in which case the following additional conditions are satisfied: rzz ðx; 0Þ 6 0
for x 2 ða; bÞ;
½wðx; 0Þ P 0 for x 2 ða; aÞ:
The first one guarantees that the crack faces in ða; bÞ are compressed and the second one excludes interpenetration of the crack faces for ða; aÞ. Parameter k0 is the real contact zone length, which can be found from the equation K1 ¼ 0, as it was shown for elastic orthotropic materials in [26]. In this case, K2 ¼ K20 is the main parameter of fracture. For real piezoceramics, the value of k0 is extremely small. But it is clear from the results of calculations in Table 1 that the value of G is almost insensitive to the variation of k in the vicinity of k0 . It means that, similar to elastic orthotropic materials [26], the ERR G is quasi-invariant with respect to k and ðG G0 Þ=G0 ¼ 0ðkÞ, where G0 is the ERR corresponding to the real contact zone length k0 . Thus, to simplify an interface crack problem solution, the ERR G can be found for the same crack but with some artificial contact zone with k 103 –102 . After that, due to the quasi-invariance phenomenon discussed above, the value of G0 can be taken equal to G with the error of 0ðkÞ. Next, we can determine K20 from a relation which follows from Eq. (33) when K1 ðk0 Þ ¼ 0
Fig. 2. The variation of the normal stress along the fixed end for various relative semi-infinite strip widths (n=b ¼ 1:0).
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sffiffiffiffiffiffiffiffiffiffiffiffiffi 4G0 K20 ¼ ðb11 < 0Þ: pb11 Figs. 2 and 3 refer to permeable crack faces. They show, respectively, the variation of the normal stress rzz along the fixed end of the semi-infinite strip for various relative strip widths and locations of the applied concentrated loading. The results in Fig. 2 are obtained for the position of the concentrated force Fz defined
Fig. 3. The variation of the normal stress along the fixed end for various locations of the applied concentrated loading (h=b ¼ 4:0).
Fig. 4. Behaviour of the shear stress along the fixed end under different electric boundary conditions of the crack.
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by n=b ¼ 1:0. Fig. 3 is obtained for the relative strip width defined by h=b ¼ 4:0. In Fig. 2 can be seen that the decrease of the strip width leads to an increase of rzz ðx; 0Þ at all points in the interval ½b; h and especially at the crack tip. The variation of the position of the concentrated force Fz in Fig. 3 can retard or enhance the stress level in the vicinity of the crack tip. Different from what is expected at first sight, closer analysis of the numerical results did not indicate that the curves do intersect in one single point. Fig. 4 shows the influence of the electric crack permeability on rxz ; ðx; 0Þ. The behaviour rxz ðx; 0Þ in the vicinity of the crack tip determines the SIF K2 . The influence of different electric permeabilities is considered: the permeable crack, the limited permeable crack with air as crack medium (er ¼ 1), and the impermeable crack. It is interesting to note that the results for all three cases do not differ essentially. Moreover in the near-crack tip zone the values of stresses for permeable and limited permeable cracks are almost the same and graphically they coincide completely. Figs. 5 and 6 show the effect of the electric permeability of the crack on the intensity factor K20 and on the energy release rate G0 . In the calculation presented here, er varies between 0 (impermeable crack) and 7.
Fig. 5. Dependence of the stress intensity factor K20 on the relative permittivity er of the crack medium for real electric permeability (graph a) and for fictitious medium (graph b).
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Fig. 6. Dependence of the energy release rate G0 on the relative permittivity er of the crack medium for real electric permeability (graph a) and for fictitious medium (graph b).
It is worth a reminder that in reality only crack media with er P 1 exist. It is seen from Figs. 5b and 6b that only cracks with a fictitious crack medium have a more pronounced effect on K20 and G0 due to electric permeability, but for a real electric permeability this effect is very small (Figs. 5a and 6a). It means that rather appropriate values of K20 and G0 can be obtained for any er P 1 by consideration of a completely permeable crack.
7. Conclusion The plane strain state of an interface crack between a rigid conductor and a piezoelectric semi-infinite strip, loaded by a concentrated force and a concentrated electric charge, is considered. Different conditions on the crack faces––permeable, impermeable and limited permeable––are considered. The boundary
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integral relations for a semi-infinite piezoelectric strip are found, and using these relations the system of singular integral equations corresponding to all three types of boundary conditions is formulated. The quasi-invariance of the energy release rate with respect to the artificial contact zone length is confirmed, and the appropriate method of fracture mechanical parameters determination which can be used for a finite sized domain, is suggested. It is found that a electromechanical field can retard or enhance the stress level in the vicinity of the crack tip, depending on the magnitude and the location of the applied concentrated loading. Therefore, optimisation of the magnitude and the location of the applied loading for the sake of decreasing the SIF at the interface crack tip can be suggested. The influence of the electrical permeability of the crack on the electromechanical fields at the crack tip and on the fracture mechanics parameters is considered. It is shown that fracture mechanics parameters of limited permeable cracks differ strongly from the associated values for the impermeable crack, but the results for permeable and limited permeable cracks are almost the same. Therefore, the often used approximation of air-filled impermeable cracks is less suitable then another actively used approximation based upon the completely permeable crack assumption.
Acknowledgements The first author (VG) gratefully acknowledges the support given to him by the Alexander von Humboldt Foundation for carrying out this work at Forschungszentrum Karlsruhe (Institute for Materials Research II).
References [1] Kamlah M. Ferroelectric and ferroelastic piezoceramics––modeling of electromechanical hysteresis phenomena. Continuum Mech Thermodyn 2001;13:219–68. [2] Deeg WF. The analysis of dislocation, crack, and inclusion problems in piezoelectric solids. PhD thesis, Stanford University, 1980. [3] Parton VZ, Kudryavtsev BA. Electromagnetoelasticity. New York: Gordon and Breach; 1988. [4] McMeeking RM. Electrostrictive stresses near crack-like flaws. J Appl Math Phys 1989;40:615–27. [5] Pak YE. Crack extension force in a piezoelectric material. J Appl Mech 1990;57:647–53. [6] Pak YE. Linear electro-elastic fracture mechanics of piezoelectric materials. Int J Fract 1992;54:79–100. [7] Sosa H. Plane problems in piezoelectric media with defects. Int J Solids Struct 1991;28:491–505. [8] Sosa H. On the fracture mechanics of piezoelectric solids. Int J Solids Struct 1991;29:2613–22. [9] Dunn ML. The effect of crack faces boundary conditions on the fracture mechanics of piezoelectric solids. Engng Fract Mech 1994;48:25–39. [10] Park SB, Sun CT. Effect of electric field on fracture of piezoelectric ceramics. Int J Fract 1995;70:203–16. [11] Gao H, Zhang TY, Tong P. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J Mech Phys Solids 1997;45:491–510. [12] Parton VZ. Fracture mechanics of piezoelectric materials. Acta Astronaut 1976;3:671–83. [13] Suo Z, Kuo CM, Barnett DM, Willis JR. Fracture mechanics for piezoelectric ceramics. J Mech Phys Solids 1992;40:739–65. [14] Beom HG, Atluri SN. Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media. Int J Fract 1996;75:163–83. [15] Gao CF, Wang MZ. Collinear permeable cracks between dissimilar piezoelectric materials. Int J Solids Struct 2000;37:4969–86. [16] Govorukha VB, Munz D, Kamlah M. On the singular integral equations approach to the interface crack problem for piezoelectric materials. Arch Mech 2000;52:247–73. [17] Ma LF, Chen YH. Weight functions for interface cracks in dissimilar anisotropic piezoelectric materials. Int J Solids Struct 2001;110:263–79. [18] Williams ML. The stress around a fault or cracks in dissimilar media. Bull Seismol Soc Am 1959;49:199–204. [19] Comninou M. The interface crack. J Appl Mech 1997;44:631–6. [20] Qin QH, Mai YW. A closed crack model for interface cracks in thermopiezoelectric materials. Int J Solids Struct 1999;36:2463–79. [21] Herrmann KP, Loboda VV. On interface crack models with contact zones situated in an anisotropic bimaterial. Arch Appl Mech 1999;69:311–35.
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[22] Herrmann KP, Loboda VV. Fracture-mechanical assessment of electrically permeable interface cracks in piezoelectric bimaterials by consideration of various contact zone models. Arch Appl Mech 2000;70:127–43. [23] Herrmann KP, Loboda VV, Govorukha VB. On contact zone models for an electrically impermeable interface crack in a piezoelectric bimaterial. Int J Fract 2001;111:203–27. [24] Hao TH, Shen ZY. A new electric boundary condition of electric fracture mechanics and its applications. Engng Fract Mech 1994;47:793–802. [25] Muskhelisvili NI. Singular integral equations. Groningen: Noordhoff; 1953. [26] Loboda VV. The problem of an orthotropic semi-infinite strip with a crack along the fixed end. Engng Fract Mech 1996;55:7–17. [27] Dunn ML, Taya M. Electroelastic field concentrations in and around inhomogeneities in piezoelectric solids. J Appl Mech 1993;61:474–5.
The interface crack problem for a piezoelectric semi-infinite strip under concentrated electromechanical loading V. Govorukha a
a,*
, M. Kamlah b, D. Munz
b
Department of Computational Mathematics, Dnepropetrovsk National University, 49050 Dnepropetrovsk, Ukraine b Forschungszentrum Karlsruhe, Institut for Materials Research II, Postfach 3640, 76021 Karlsruhe, Germany Received 19 May 2003; received in revised form 15 October 2003; accepted 2 December 2003
Abstract A plane strain problem for an interface crack along the fixed edge of a piezoelectric semi-infinite strip under concentrated electromechanical loading is examined. The model of an interface crack with an artificial contact zone near its tips is used. By using Fourier transforms, boundary integral relations are derived. By means of these relations, the system of singular integral equations for several electric boundary conditions are formulated and solved numerically. The stress intensity factors at the crack tip and the energy release rate are calculated. Influence of the permeability of the crack on electric and mechanical fields near the crack tip is considered. 2004 Elsevier Ltd. All rights reserved. Keywords: Piezoelectric material; Interface crack; Contact zone; Permittivity of crack medium
1. Introduction Piezoelectric materials exhibit an interesting phenomenon in that they produce an electric field when deformed and undergo deformation when subjected to an electric field. An overview over microscopic and phenomenological approaches related to electromechanical coupling phenomena is given in [1]. Due to this intrinsic electromechanical coupling behaviour, piezoelectric materials have been widely used in technology as sensors and actuators. On the other hand piezoelectric materials are very brittle and susceptible to fracture. Therefore, in order to predict the performance and the integrity of piezoelectric devices, it is important that the behaviour of various defects are analyzed and studied under electrical and mechanical fields. A wealth of theoretical results, in the field of piezoelectric fracture, has been presented in [2–11]. They have solved basic problems for cracks in homogeneous piezoelectric materials and introduced some fracture criteria for piezoelectric ceramics. However, the problem of an interface crack in a piezoelectric bimaterial, *
Corresponding author. E-mail address: [email protected] (V. Govorukha).
0013-7944/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2003.12.005
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Nomenclature rij stress components sij strain components Ei electric field components Di electric displacement components u, w components of the displacement vector u electric potential fi body force density q body charge density cij elastic constants eij dielectric constants eij piezoelectric constants ðn; gÞ coordinates of the concentrated electromechanical loading Fx , Fz components of the concentrated force q0 magnitude of the concentrated electric charge 2d thickness of the interface crack ea permittivity of the medium inside the interface crack er relative permittivity of the medium 2h width of the piezoelectric semi-infinite strip 2b length of the interface crack a < jxj < b frictionless contact zones at the tip of the crack k ¼ ðb aÞ=2b relative length of the contact zone k0 real contact zone length Ki stress intensity factors G energy release rate K20 ¼ K2 ðk0 Þ stress intensity factor K2 corresponding to the real contact zone length G0 ¼ Gðk0 Þ energy release rate corresponding to the real contact zone length All other symbols are mathematical constants or represent abbreviations for mathematical expressions and do not denote fundamental physical quantities.
in spite of its importance, has not obtained due attention in the literature because of its complexity. The first attempt to analyze piezoelectric interface crack problems from a fracture mechanics point of view appears to be that of Parton [12], who analyzed a finite crack at the interface of a piezoelectric material and a conductor subjected to a far-field uniform tension. Later, Suo et al. [13] examined the problem of an interface crack between dissimilar anisotropic piezoelectric media and the dependence of singularities at the tips of an interface crack with respect to different electrical conditions. A new type of singularity has been found in these papers. Beom and Atluri [14] investigated a complete form of stress and electric displacement field in the vicinity of an interface crack in dissimilar anisotropic piezoelectric materials, and defined a new set of stress and electric displacement intensity factors. The generalized two-dimensional problem of collinear interfacial cracks between two dissimilar piezoelectric media subjected to piecewise uniform loads at infinity has been studied by Gao and Wang [15]. The singular integral equations approach to the interface crack problem for piezoelectric materials has been studied by Govorukha et al. [16]. Weight function analysis for interface cracks in dissimilar anisotropic piezoelectric materials has been proposed and investigated by Ma and Chen [17].
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It should be noted that in most of the above-mentioned investigations were performed in the frame of the classical interface crack model [18], initially assuming that the interface crack is completely opened and leading to an oscillating singularity at the crack tips. But along with the classical interface crack model, the contact zone model [19] was developed for a crack between two non-piezoelectric materials. This model was applied to interface cracks in thermo-piezoelectric materials by Qin and Mai [20]. Besides, the usefulness of a new interface crack model based on an artificial contact zone was demonstrated by Herrmann and Loboda [21] and in particular cases, classical model and contact zone model, were found. On the basis of this model a detailed analytical investigation of interface crack between two piezoelectric semiinfinite planes with respect to different electrical conditions has been performed in recent investigations [22,23]. In the theoretical analyses of the crack problem in piezoelectric materials, the researchers assume the crack surfaces to be stress-free, but they have different opinions about the electrical boundary condition at the crack surfaces. As the dielectric constant of air or the medium between the crack faces is very small compared to that of the piezoelectric material Deeg [2], Pak [5] have assumed crack surfaces to be free of surface charge. On the other hand Parton [12] has assumed that although the magnitude of the normal electrical displacement component at the crack face is very small, the electric displacement is continuous across the crack faces. Using the analogy of a capacitor filled by a medium, Hao and Shen [24] used a boundary condition in which the electrical permeability of air in the crack gap is considered. Detailed discussion of this topic is presented in the following sections. In most of the above studies the mechanical and electrical loading is chosen to be acting upon the crack faces or at infinity. But in some practical cases, a mechanical or a electrical load can be applied at the internal points of the domain. To the best of our knowledge, correspondent results for piezoelectric interface crack problems have not been obtained yet. In this paper the interface problem for a crack along the fixed end of an piezoelectric semi-infinite strip is investigated with taking into account the permittivity of the crack medium. The model of a crack with an artificial contact zone near the tips is utilized. The problem is reduced to a system of singular integral equations corresponding to different electrical conditions. The numerical solution of these systems of equations gives the possibility to find main fracture characteristics for all the considered cases.
2. Basic equations The constitutive equations for the piezoelectric materials can be written as [3] rij ¼ cijkl skl ekij Ek ;
ð1Þ
Di ¼ eikl skl þ eik Ek ;
ð2Þ
where rij , skl , Ek and Di , are stress, strain, electric field and electric displacement, respectively. cijkl , eik and eikl are the elastic constants, the dielectric constants and the piezoelectric constants tensor of the material, respectively. The strain tensor is related to the displacement vector fui ; i ¼ 1; 2; 3g by the equation sij ¼ ðui;j þ uj;i Þ=2;
ð3Þ
where a comma indicates the partial derivative. The electric field strength is related to the electric potential u by the equation Ei ¼
ou : oxi
ð4Þ
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The governing field equations are given by rij;j ¼ fi ;
ð5Þ
Di;i ¼ q;
ð6Þ
where fi and q are the body force and body charge, respectively. Poled piezoelectric materials like barium titanate (BaTiO3 ) and lead zirconate titanate (PZT) are transversely isotropic materials with z in the poling direction. For the latter, a plane strain response is considered and a Cartesian coordinate system ðx; y; zÞ is positioned in such a manner that the plane deformation is orthogonal to the y-axis. Eqs. (1)–(6) are written for this coordinate system. Then Eqs. (3) and (4) are substituted into Eqs. (1) and (2) and the result is put into Eqs. (5) and (6). The governing differential equations now become o2 u o2 u o2 w o2 u þ ðe ¼ Fx dðx nÞdðz gÞ; þ c þ ðc þ c Þ þ e Þ 44 13 44 31 15 ox2 oz2 ox oz ox oz o2 u o2 w o2 w o2 u o2 u þ c44 2 þ c33 2 þ e33 2 þ e15 2 ¼ Fz dðx nÞdðz gÞ; ðc13 þ c44 Þ ox oz ox oz oz ox o2 u o2 w o2 w o2 u o2 u þ e15 2 þ e33 2 e11 2 e33 2 ¼ q0 dðx nÞdðz gÞ; ðc13 þ c44 Þ ox oz ox oz ox oz
c11
ð7Þ
where fFx ; Fz g are components of the concentrated force and q0 is the magnitude of the concentrated electric charge, which are applied at point ðn; gÞ. dðxÞ is the Dirac-delta function. In this system the displacement components uðx; zÞ, wðx; zÞ, and the electric potential uðx; zÞ are the unknown functions.
3. The electrical boundary conditions on the faces of the crack It has already been noted that several electric boundary conditions were proposed in literature. They differ in the shielding strength of the electric displacement at the crack. A useful approach to the formulation of such conditions was suggested in [3]. The boundary conditions were formulated under the assumption that the surfaces of contacting bodies are separated by a thin intermediate layer describable in terms of certain reduced physical parameters. If the parameters are considered to be constant, the conditions for the potential u and the electric displacement normal component Dz , on the physical surface of the transition layer read as ex
o2 þ ðu þ u Þ 2ðDþ z Dz Þ ¼ 0; ox2
ð8Þ
ex
o2 þ þ ðu u Þ 6ðDþ z Dz Þ 12ez ðu u Þ ¼ 0: ox2
ð9Þ
Superscripts ‘‘+’’ and ‘‘)’’ denote the upper and lower crack surfaces, respectively. It is seen that the electrical boundary conditions on the surface of the layer are characterized by the two parameters, ex ¼ ea 2d;
ez ¼ ea =ð2dÞ;
ð10Þ
where 2d is the thickness and ea is the permittivity of the layer, respectively. Eqs. (8) and (9) may be used for deriving the electrical boundary conditions at the faces of a crack in a piezoelectric medium. Some special cases are as follows:
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(i) If ex ! 0 and ez ! 1, Eqs. (8) and (9) lead to the permeable boundary condition Dþ z ¼ Dz ;
uþ ¼ u ;
ð11Þ
which does not shield the electric displacement at all. (ii) When ex ¼ ez ¼ 0, we obtain the impermeable boundary condition Dþ z ¼ Dz ¼ 0;
ð12Þ
which is supposed to shield the electric displacement completely. (iii) Letting ex ¼ 0 and ez 6¼ 0 yields the boundary condition, in which the shielding of the crack depends on the permittivity ea of the medium inside the crack, Dþ z ¼ Dz ¼ Dz ;
uþ u ¼
1 Dz : ez
ð13Þ
This boundary condition is called limited permeable. It is worth noting that the boundary condition (13) is related to the boundary condition [24]. In the formulation used here, the thickness d is a constant. However, this restriction appeared to be necessary, because the solution of the problems considered in the following sections, seems to be possible only on the basis of the form (13) of the limited permeable boundary condition with constant thickness. The thickness d will be assigned a physically reasonable value in the context of the solution of the problem in Section 5.3. In this paper the medium is defined by its relative permittivity er , which is the ratio between the permittivity ea and the permittivity of vacuum, e0 ¼ 8:85 1012 A s/V m: ea ¼ er e0 :
ð14Þ
For air and vacuum, er ¼ 1. There is no physical medium with er < 1. The permeable and impermeable boundary conditions may be considered as the extreme cases of the limited permeable boundary condition.
4. Formulation of boundary integral relations for a semi-infinite piezoelectric strip Consider a plane strain state in the plane ðx; zÞ for a piezoelectric semi-infinite strip shown in Fig. 1. Let in the points Aðn; gÞ and A0 ðn; gÞ concentrated forces and electric charges be symmetrically applied. The boundary conditions on the sides x ¼ h of the semi-infinite strip are given in the form uðh; zÞ ¼ 0;
ow ðh; zÞ ¼ 0; ox
ou ðh; zÞ ¼ 0: ox
ð15Þ
The electromechanical behaviour of the considered material under concentrated loads is fully described by the governing equations (7). In order to solve the differential equations (7) we introduce the finite Fourier transforms uðp; zÞ ¼
Z
h
uðx; zÞ sinðxpxÞ dx; 0
ðp; zÞ ¼ w
Z
h
wðx; zÞ cosðxpxÞ dx;
ðp; zÞ ¼ u
0
Z
h
uðx; zÞ cosðxpxÞ dx 0
ð16Þ
are governed by the ordinary differential equations and u in which case u, w
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Fig. 1. The piezoelectric semi-infinite strip of width 2h under concentrated electromechanical loading. The ranges ða; bÞ represents the frictionless contact zone. The location of the concentrated force fFx ; Fz g and electric charge q0 is ðn; gÞ.
d2 d w d u u xpðe31 þ e15 Þ ¼ Fx sinðxpnÞdðz gÞ; xpðc13 þ c44 Þ dz dz dz2 d u d2 w d2 u þ e33 2 ¼ Fz cosðvpnÞdðz gÞ; þ c33 2 x2 p2 e15 u xpðc13 þ c44 Þ x2 p2 c44 w dz dz dz 2 2 d u d d w u e33 þ e33 2 þ x2 p2 e11 u ¼ q0 cosðxpnÞdðz gÞ; xpðe15 þ e31 Þ x2 p2 e15 w dz dz dz x2 p2 c11 u þ c44
ð17Þ
where x ¼ p=h (we have taken into account that due to symmetry uð0; zÞ ¼ 0, owð0; zÞ=ox ¼ 0, ouð0; zÞ=ox ¼ 0). Applying further Fourier transforms with respect to the z-coordinate, Z 1 Z 1 Z 1 ^ ðp; tÞ ¼ ðp; zÞ sinðtzÞ dz; ^ ðp; tÞ ¼ ðp; zÞ sinðtzÞ dz; ^ uðp; zÞ cosðtzÞ dz; w w u u uðp; tÞ ¼ 0
0
0
^: ^ and u to Eq. (17), we arrive at the following algebraic system with respect to ^u, w ðx2 p2 c11 þ c44 t2 Þ^ u þ xptðc13 þ c44 Þ^ w þ xptðe31 þ e15 Þ^ u ¼ q1 þ xpc13 Q2 þ xpe31 Q3 þ Fx sinðxpnÞ cosðtgÞ; u þ ðx2 p2 c44 þ c33 t2 Þ^ w þ ðx2 p2 e15 þ t2 e33 Þ^ u ¼ tc33 Q2 þ te33 Q3 þ Fz cosðxpnÞ sinðtgÞ; xptðc13 þ c44 Þ^ xptðe15 þ e31 Þ^ u þ ðx2 p2 e15 þ e33 t2 Þ^ w ðx2 p2 e11 þ t2 e33 Þ^ u ¼ te33 Q2 te33 Q3 q0 cosðxpnÞ sinðtgÞ:
ð18Þ
Here, Q2 ðpÞ ¼
Z
h
Q2 ðxÞ cosðxpxÞ dx;
Q3 ðpÞ ¼
0
Z
h
Q3 ðxÞ cosðxpxÞ dx; 0
and q1 ðxÞ ¼ rxz ðx; 0Þ;
Q2 ðxÞ ¼ wðx; 0Þ;
Q3 ðxÞ ¼ uðx; 0Þ:
q1 ðpÞ ¼
Z
h
q1 ðxÞ sinðxpxÞ dx 0
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The form of the solution of this system depends on the roots of the characteristic equation At6 þ Bt4 p2 x2 þ Ct2 p4 x4 þ Dp6 x6 ¼ 0;
ð19Þ
where A ¼ c44 ðc33 e33 þ e233 Þ; B ¼ 2e33 ðc13 þ c44 Þðe31 þ e15 Þ c11 c33 e33 c244 e33 c44 c33 e11 c33 ðe15 þ e31 Þ2 2
2e33 e15 c44 e233 c11 þ e33 ðc13 þ c44 Þ ; C ¼ 2e15 ðc13 þ c44 Þðe31 þ e15 Þ c11 c44 e33 c11 c33 e11 c244 e11 c44 ðe15 þ e31 Þ
2
e215 c44 2e15 e33 c11 þ e11 ðc13 þ c44 Þ2 ; D ¼ c11 ðc44 e11 þ e215 Þ: Analysis of Eq. (19) shows that for realistic values of the material constants of the considered class of materials, this equation has two real roots k1 and two pairs of complex conjugate roots ðd ihÞ [3]. Using the inverse Fourier transforms for the solution of the system (18) leads to the following expressions: 3 Z h ou 1 X o~u ðx; zÞ ¼ f2c1i R1 ðx; z; tÞ þ c2i R2 ðx; z; tÞ þ c3i R3 ðx; z; tÞgqi ðtÞ dt þ ðx; zÞ; ox 2h i¼1 0 ox 3 Z h ow 1 X o~ w ðx; zÞ ¼ ðx; zÞ; f2q1i R1 ðx; z; tÞ þ q2i R2 ðx; z; tÞ þ q3i R3 ðx; z; tÞgqi ðtÞ dt þ oz 2h i¼1 0 oz
ð20Þ
3 Z h ou 1 X o~ u ðx; zÞ ¼ ðx; zÞ; f2v1i R1 ðx; z; tÞ þ v2i R2 ðx; z; tÞ þ c3i R3 ðx; z; tÞgqi ðtÞ dt þ oz 2h i¼1 0 oz
where q2 ðtÞ ¼ Q02 ðtÞ; R1 ðx; z; tÞ ¼
R2 ðx; z; tÞ ¼
q3 ðtÞ ¼ Q03 ðtÞ;
sin½xðt þ xÞ sin½xðt xÞ þ ; cosh½xa1 z cos½xðt þ xÞ cosh½xa1 z cos½xðt xÞ 2 X n¼1
R3 ðx; z; tÞ ¼
2 X
n n sin½xðt þ x þ ð1Þ a5 zÞ sin½xðt x þ ð1Þ a5 zÞ þ ; n n cosh½xa4 z cos½xðt þ x þ ð1Þ a5 zÞ cosh½xa4 z cos½xðt x þ ð1Þ a5 zÞ
ð1Þ
nþ1
n¼1
a21 ¼ jk1 j;
a22 ¼ jdj;
sinh½xa4 z sinh½xa4 z þ ; n n cosh½xa4 z cos½xðt þ x þ ð1Þ a5 zÞ cosh½xa4 z cos½xðt x þ ð1Þ a5 zÞ
a23 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 2 þ h2 ;
a24 ¼ ða23 þ a22 Þ=2;
a25 ¼ ða23 a22 Þ=2:
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The following values are defined by the external load: 2 o~ u 1 X ðx; zÞ ¼ ½f~c Li ðx; z; n; gÞ þ ~c12 Xi ðx; z; n; gÞ þ ~c13 Yi ðx; z; n; gÞgFx ox 4h i¼1 11
þ f~c21 Ni ðx; z; n; gÞ þ ~c22 Xi ðx; z; n; gÞ þ ~c23 Yi ðx; z; n; gÞgFz þ f~c31 Ni ðx; z; n; gÞ þ ~c32 Xi ðx; z; n; gÞ þ ~c33 Yi ðx; z; n; gÞgq0 ; 2 o~ w 1 X ~12 Xi ðx; z; n; gÞ þ q ~13 Yi ðx; z; n; gÞgFx ðx; zÞ ¼ ½f~ q11 Li ðx; z; n; gÞ þ q oz 4h i¼1
~22 Xi ðx; z; n; gÞ þ q ~23 Yi ðx; z; n; gÞgFz þ f~ q21 Ni ðx; z; n; gÞ þ q
ð21Þ
~32 Xi ðx; z; n; gÞ þ q ~33 Yi ðx; z; n; gÞgq0 ; þ f~ q31 Ni ðx; z; n; gÞ þ q 2 o~ u 1 X ðx; zÞ ¼ ½f~ v Li ðx; z; n; gÞ þ ~ v12 Xi ðx; z; n; gÞ þ ~v13 Yi ðx; z; n; gÞgFx oz 4h i¼1 11
þ f~ v21 Ni ðx; z; n; gÞ þ ~ v22 Xi ðx; z; n; gÞ þ ~v23 Yi ðx; z; n; gÞgFz þ f~ v31 Ni ðx; z; n; gÞ þ ~ v32 Xi ðx; z; n; gÞ þ ~v33 Yi ðx; z; n; gÞgq0 ; where Li ðx; z; n; gÞ ¼
sin½xðn þ xÞ sin½xðn xÞ þ ; i cosh½xa1 ðz þ ð1Þ gÞ cos½xðn þ xÞ cosh½xa1 ðz þ ð1Þi gÞ cos½xðn xÞ
sinh½xa1 ðz þ ð1Þi gÞ sinh½xa1 ðz þ ð1Þi gÞ þ ; i i cosh½xa1 ðz þ ð1Þ gÞ cos½xðn þ xÞ cosh½xa1 ðz þ ð1Þ gÞ cos½xðn xÞ ( 2 X sin½xðn þ x þ ð1Þn a5 ðz þ ð1Þi gÞÞ Xi ðx; z; n; lÞ ¼ i n i cosh½xa4 ðz þ ð1Þ gÞ cos½xðn þ x þ ð1Þ a5 ðz þ ð1Þ gÞÞ n¼1 ) n i sin½xðn x þ ð1Þ a5 ðz þ ð1Þ gÞÞ þ ; cosh½xa4 ðz þ ð1Þi gÞ cos½xðn x þ ð1Þn a5 ðz þ ð1Þi gÞÞ Ni ðx; z; n; gÞ ¼
Yi ðx; z; n; gÞ ¼
2 X
(
i
sinh½xa4 ðz þ ð1Þ gÞ i cosh½xa ðz þ ð1Þ gÞ cos½xðn þ x þ ð1Þn a5 ðz þ ð1Þi gÞÞ 4 n¼1 ) i sinh½xa4 ðz þ ð1Þ gÞ þ : i n i cosh½xa4 ðz þ ð1Þ gÞ cos½xðn x þ ð1Þ a5 ðz þ ð1Þ gÞÞ
~ij , ~ vij (i; j ¼ 1; 2; 3) are defined by the electric and elastic parameters of the The real constants cij , qij , vij , ~cij , q material. Their actual expressions are quite lengthy and are omitted because they do not introduce any relevant information in the final results. Expressions for the entries in o~u=ox, o~ w=oz and o~ u=oz shown above are valid for z P g. For z 6 g, these expressions are expressible by interchanging z with g. Extracting the singular components of the integrals in (20), we obtain the following formulas for strain, stress and electric displacement on edge of the semi-infinite strip: Z h 3 1X 1 u0 ðx; 0Þ ¼ b1i ð22Þ þ Mðx; tÞ qi ðtÞ dt þ ~u0 ðx; 0Þ; p i¼1 h t x
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rzz ðx; 0Þ ¼
Z h 3 1X 1 ~zz ðx; 0Þ; b2i þ Mðx; tÞ qi ðtÞ dt þ r p i¼1 h t x
ð23Þ
Dz ðx; 0Þ ¼
Z h 3 1X 1 ~ z ðx; 0Þ; b3i þ Mðx; tÞ qi ðtÞ dt þ D p i¼1 h t x
ð24Þ
where u0 ðx; 0Þ ¼
ou ðx; 0Þ; ox
b1i ¼ c1i þ c2i ; b2i ¼ c13 ðc1i þ c2i Þ þ c33 ðq1i þ q2i Þ þ e33 ðv1i þ v2i Þ; b3i ¼ e31 ðc1i þ c2i Þ þ e33 ðq1i þ q2i Þ þ e33 ðv1i þ v2i Þ; ~zz ðx; 0Þ ¼ c13 r
o~ u o~ w o~ u ðx; 0Þ þ c33 ðx; 0Þ þ e33 ðx; 0Þ; ox oz oz
u o~ w o~ u ~ z ðx; 0Þ ¼ e13 o~ D ðx; 0Þ þ e33 ðx; 0Þ e33 ðx; 0Þ; ox oz oz x sin½xðt þ xÞ sin½xðt xÞ 2 Mðx; tÞ ¼ þ ; 2 1 cos½xðt þ xÞ 1 cos½xðt xÞ xðt xÞ
i ¼ 1; 2; 3:
In the last expressions the function Mðx; tÞ 2 H , where H is the class of H€ older functions [25]. The boundary integral relations (22)–(24) play an important role in the following analysis because by means of these relations the singular integral equations for various boundary conditions at the interface can be formulated.
5. An interface crack under concentrated electromechanical loadings 5.1. Solution of the permeable crack problem Suppose that a semi-infinite piezoelectric strip is perfectly bounded at the edge z ¼ 0, b < jxj < h to a rigid conductor and an interface crack is situated along the interval z ¼ 0, jxj < b. In this section, we consider the permeable crack problem. The boundary conditions are the following: uðh; zÞ ¼ 0; rxz ðx; 0Þ ¼ 0;
ow ou ðh; zÞ ¼ 0; ðh; zÞ ¼ 0; ox ox rzz ðx; 0Þ ¼ 0; jxj < b;
uðx; 0Þ ¼ 0;
wðx; 0Þ ¼ 0;
uðx; 0Þ ¼ 0;
jxj < h:
ð25Þ
b < jxj < h;
According to (25) one can set in Eqs. (22)–(24) q1 ðxÞ ¼ 0 for jxj < b, q2 ðxÞ ¼ 0 for b < jxj < h and q3 ðxÞ ¼ 0. Also making use of the symmetry properties of these unknown functions and satisfying further the equations u0 ðx; 0Þ ¼ 0 and rzz ðx; 0Þ ¼ 0, we obtain the following system of singular integral equations (SIE):
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Z bi1 b
h
Z b 1 1 1 þ M ðx; tÞ q1 ðtÞ dt þ bi2 þ þ Mðx; tÞ q2 ðtÞ dt ¼ ri ; tx tþx b t x
ð26Þ
where x 2 ðb; hÞ for i ¼ 1, and x 2 ðb; bÞ for i ¼ 2, M ðx; tÞ ¼ Mðx; tÞ Mðx; tÞ, r1 ¼ p~u0 ðx; 0Þ, r2 ¼ p~ rzz ðx; 0Þ. In this case we obtain singularities [18], which imply oscillations in the stress field near the tips of the crack, resulting in the unrealistic phenomenon of interpenetrating crack faces, and leading to difficulties in the numerical analysis for the determination of the fracture parameters. For this reason, similarly to [19] we introduce frictionless contact zones a < jxj < b to avoid the oscillating singularity, where the position of the point a is arbitrary for the time being. For such an arbitrary position of the point a, we have an artificial contact zone model, which is not physically justified, but from this model the specific value of a for the real contact zone length can be found. The boundary conditions at z ¼ 0 for this model are the following: rxz ðx; 0Þ ¼ 0; rxz ðx; 0Þ ¼ 0;
rzz ðx; 0Þ ¼ 0; wðx; 0Þ ¼ 0;
uðx; 0Þ ¼ 0; wðx; 0Þ ¼ 0; uðx; 0Þ ¼ 0; jxj < h:
jxj < a; a < jxj < b;
ð27Þ
b < jxj < h;
In this case q1 ðxÞ ¼ 0 for jxj < b, q2 ðxÞ ¼ 0 for a < jxj < h and q3 ðxÞ ¼ 0, and we arrive at the very same system of SIE (26), in which now x 2 ðb; hÞ for i ¼ 1, and x 2 ða; aÞ for i ¼ 2. Additional conditions must be satisfied to ensure that the displacements are single-valued: Z a Z b ouðx; 0Þ owðx; 0Þ dx ¼ 0; dx ¼ 0: ox ox b a When making use of relation (22), these conditions can be written in the form Z h Z a 1 cos½xðt þ bÞ 1 cos½xðt þ bÞ q2 ðtÞ dt q1 ðtÞ dt þ b12 2b11 ln ln 1 cos½xðt bÞ 1 cos½xðt bÞ b a Z a q2 ðtÞ dt ¼ 0: ¼ 2p~ uðb; 0Þ
ð28Þ
a
Due to the absence of the oscillating singularity in this case, the unknown functions qi ðtÞ (i ¼ 1; 2) can be represented in the form q1 ðtÞ q1 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðh tÞðt bÞ
q2 ðtÞ ffi; q2 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi a2 t 2
qi ðtÞ 2 H :
Next we introduce the stress intensity factors (SIF) for the right crack tip as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K1 ¼ lim 2ðx aÞrzz ðx; 0Þ; 2ðx bÞrxz ðx; 0Þ: K2 ¼ lim x!aþ0
x!bþ0
Due to (23), (29) and the results [25] for Cauchy type integral behaviour, we can write rffiffiffiffiffiffiffiffiffiffiffi b22 q2 ðaÞ 2 K1 ¼ pffiffiffi ; K2 ¼ q ðbÞ: hb 1 a
ð29Þ
ð30Þ
ð31Þ
The energy release rate (ERR) G for the right crack tip can be presented by the virtual work integral [3] Z aþDl Z bþDl 1 rzz ðx; 0Þwðx þ Dl; 0Þ dx þ rxz ðx; 0Þuðx þ Dl; 0Þ dx : ð32Þ G ¼ lim Dl!0 2 Dl a b
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The integrals in the last formula can be computed from the asymptotic behaviour of displacements and stresses near the singular points a and b. Using the asymptotic behaviour of q2 ðxÞ for x ! a 0 we find from the second of relations (16) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ða xÞ q2 ðaÞ: wðx; 0Þjx!a0 ¼ a Employing Cauchy type properties in Eq. (23) gives q2 ðaÞ rzz ðx; 0Þjx!aþ0 ¼ b22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2aðx aÞ In a similar way it can be shown that rffiffiffiffiffiffiffiffiffiffiffi bx uðx; 0Þjx!b0 ¼ 2b11 q ðbÞ; hb 1 q1 ðbÞ ffi: rxz ðx; 0Þjx!bþ0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðh bÞðx bÞ Substituting the last formulas into (32) and taking into account that Z Dl rffiffiffiffiffiffiffiffiffiffiffiffiffi Dl x p Dl dx ¼ ; x 2 0 we obtain the following formula: p b22 2 2b11 2 G¼ q2 ðaÞ q1 ðbÞ : 4 a hb
ð33Þ
5.2. Solution of the impermeable crack problem Consider now the impermeable crack problem. The boundary conditions for this case are the following: uðh; zÞ ¼ 0; rxz ðx; 0Þ ¼ 0; uðx; 0Þ ¼ 0; Dz ðx; 0Þ ¼ 0;
ow ðh; zÞ ¼ 0; ox rzz ðx; 0Þ ¼ 0; wðx; 0Þ ¼ 0; jxj < b;
ou ðh; zÞ ¼ 0; ox jxj < b;
ð34Þ
b < jxj < h; uðx; 0Þ ¼ 0;
b < jxj < h:
In this case one can set in Eqs. (22)–(24) q1 ðxÞ ¼ 0 (jxj < b), q2 ðxÞ ¼ 0 (b < jxj < h) and q3 ðxÞ ¼ 0 (b < jxj < h). Further, we arrive at the following system of three SIEs: Z h Z b 3 X 1 1 1 þ M ðx; tÞ q1 ðtÞ dt þ bi1 bik ð35Þ þ þ Mðx; tÞ qk ðtÞ dt ¼ ri : tx tþx b b t x k¼2 e z ðx; 0Þ, x 2 ðb; hÞ for i ¼ 1, and x 2 ðb; bÞ for i ¼ 2; 3. Here, r3 ¼ p D The analysis of this system [16] showed that in the case of impermeable crack faces of a piezoelectric composite both the oscillating singularity and the real singularity of a power type appears. To avoid the oscillating singularity similar to the case of the permeable faces, we may introduce the frictionless contact zones a < jxj < b at the tips of the crack.
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The system of singular integral equations in this case has the form Z h Z a 3 X 1 1 1 þ M ðx; tÞ q1 ðtÞ dt þ bik bi1 þ þ Mðx; tÞ qk ðtÞ dt ¼ ri ; tx tþx b a t x k¼2 where x 2 ðb; hÞ for i ¼ 1, and x 2 ða; aÞ for i ¼ 2; 3. Additional conditions take the following form: Z h Z a 3 X 1 cos½xðt þ bÞ 1 cos½xðt þ bÞ q1 ðtÞ dt þ ln b ln 2b11 1k 1 cos½xðt bÞ qk ðtÞ dt ¼ 2p~uðb; 0Þ 1 cos½xðt bÞ b a k¼2 Z a qk ðtÞ dt ¼ 0 ðk ¼ 2; 3Þ:
ð36Þ
ð37Þ
a
We present the solution of the system (36) in the form q1 ðtÞ q1 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðh tÞðt bÞ
qk ðtÞ ffi qk ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi a2 t 2
ðk ¼ 2; 3Þ:
The SIFÕs K1 , K2 can be found by the same method as before and have the form rffiffiffiffiffiffiffiffiffiffiffi 1 2 K1 ¼ pffiffiffi ½b22 q2 ðaÞ þ b23 q3 ðaÞ; K2 ¼ q ðbÞ: hb 1 a
ð38Þ
ð39Þ
The ERR was found by the virtual work integral Z aþDl Z bþDl 1 rzz ðx; 0Þwðx þ Dl; 0Þ dx þ rxz ðx; 0Þuðx þ Dl; 0Þ dx G ¼ lim Dl!0 2 Dl a b Z aþDl þ Dz ðx; 0Þuðx þ Dl; 0Þ dx a
and was similarly to (32) evaluated in the form p 2ab11 2 2 2 b22 q2 ðaÞ þ b33 q3 ðaÞ G¼ q ðbÞ þ ðb23 þ b32 Þq2 ðaÞq3 ðaÞ : 4a hb 1
ð40Þ
5.3. Solution of the limited permeable crack problem The permeable and impermeable boundary conditions at the crack faces of non-conducting cracks in piezoelectric materials are usually used in literature. But both variants are too one-sided to model such cracks realistically. Instead, the electric permeability of the crack medium should be taken into account. That is why we will assume next that the crack faces are limited permeable. The boundary conditions for this crack with an artificial contact zone are the following: uðh; zÞ ¼ 0;
ow ðh; zÞ ¼ 0; ox
rxz ðx; 0Þ ¼ 0;
rzz ðx; 0Þ ¼ 0;
rxz ðx; 0Þ ¼ 0;
wðx; 0Þ ¼ 0;
uðx; 0Þ ¼ 0; Dz ðx; 0Þ ¼
wðx; 0Þ ¼ 0;
ez uðx; 0Þ;
ou ðh; zÞ ¼ 0; ox jxj < a; ð41Þ
a < jxj < b; b < jxj < h;
jxj < a;
uðx; 0Þ ¼ 0;
a < jxj < h:
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Taking into account that in this case q1 ðxÞ ¼ 0 (jxj < b), q2 ðxÞ ¼ 0 (a < jxj < h) and q3 ðxÞ ¼ 0 (a < jxj < h) and satisfying the remaining boundary conditions, we obtain the system Z h Z a 3 X 1 1 1 þ M ðx; tÞ q1 ðtÞ dt þ bik bi1 þ þ Mðx; tÞ qk ðtÞ dt ¼ ri ði ¼ 1; 2Þ; tx tþx b a t x k¼2 Z h Z Z x 3 a X 1 1 1 þ M ðx; tÞ q1 ðtÞ dt þ b3k q3 ðtÞ dt b31 þ þ Mðx; tÞ qk ðtÞ dt ¼ r3 pez tx tþx b a t x a k¼2 ð42Þ of SIEs, where x 2 ðb; hÞ for i ¼ 1, and x 2 ða; aÞ for i ¼ 2; 3. Additional conditions are taken in the form (37). The solution of the system (42) was found in the form of (38) and in order to realize the non-linear boundary conditions, an iteration scheme ne o a fez gkþ1 ¼ ð43Þ 2d k has been developed. The thickness d in the boundary condition (13) is assigned the mean crack opening: Z a 1 wðx; 0Þ dx: d¼ 2a a The iteration scheme starts with an impermeable crack assumption (iteration number 0), which gives a crack opening displacement and the right-hand side of the iteration equation (43) can be evaluated. The new field problem is solved again, leading to a new crack opening displacement, which will be once more inserted into the right-hand side of the iteration equation, and so on. The SIFs K1 and K2 have the same sense here as in (39). The ERR is defined by Z aþDl Z bþDl 1 rzz ðx; 0Þwðx þ Dl; 0Þ dx þ rxz ðx; 0Þuðx þ Dl; 0Þ dx G ¼ lim Dl!0 2 Dl a b Z aþDl Z aþDl Dz ðx; 0Þuðx þ Dl; 0Þ dx Dz ðx þ DlÞuðx þ Dl; 0Þ dx þ a
a
and can be found by the same method as earlier in the form p 2b33 2 2ab11 2 2b32 2 b22 q2 ðaÞ þ G¼ q ðaÞ q ðbÞ þ b23 þ q2 ðaÞq3 ðaÞ : 4a p 3 hb 1 p
ð44Þ
Here, uðx; 0Þ is defined as Z x uðx; 0Þ ¼ q3 ðtÞ dt: a
6. Numerical results and discussion A numerical solution of the system SIE has been obtained by the method based on the Gauss–Chebyshev quadrature rule, which was described in detail in [26]. The piezoceramic BaTiO3 is considered for the numerical analyses and its material constants are given in [27]. The results of the calculation of the SIFs K1 , K2 and ERR G for permeable crack faces are given in Table 1. The obtained results are valid in the mathematical sense for any values of the parameter
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Table 1 Dependence of the stress intensity factors K1 , K2 and the energy release rate G on the relative contact zone length k for permeable crack faces pffiffiffi pffiffiffi K2 =ðFz bÞ G=ðFz bÞ 2k K1 =ðFz bÞ 102 103 104
0.261 0.256 0.251
k¼
)0.0569 )0.0811 )0.0919
0.128 · 1011 0.129 · 1011 0.129 · 1011
ba : 2b
However, this solution is physically correct only for the particular case k ¼ k0 , in which case the following additional conditions are satisfied: rzz ðx; 0Þ 6 0
for x 2 ða; bÞ;
½wðx; 0Þ P 0 for x 2 ða; aÞ:
The first one guarantees that the crack faces in ða; bÞ are compressed and the second one excludes interpenetration of the crack faces for ða; aÞ. Parameter k0 is the real contact zone length, which can be found from the equation K1 ¼ 0, as it was shown for elastic orthotropic materials in [26]. In this case, K2 ¼ K20 is the main parameter of fracture. For real piezoceramics, the value of k0 is extremely small. But it is clear from the results of calculations in Table 1 that the value of G is almost insensitive to the variation of k in the vicinity of k0 . It means that, similar to elastic orthotropic materials [26], the ERR G is quasi-invariant with respect to k and ðG G0 Þ=G0 ¼ 0ðkÞ, where G0 is the ERR corresponding to the real contact zone length k0 . Thus, to simplify an interface crack problem solution, the ERR G can be found for the same crack but with some artificial contact zone with k 103 –102 . After that, due to the quasi-invariance phenomenon discussed above, the value of G0 can be taken equal to G with the error of 0ðkÞ. Next, we can determine K20 from a relation which follows from Eq. (33) when K1 ðk0 Þ ¼ 0
Fig. 2. The variation of the normal stress along the fixed end for various relative semi-infinite strip widths (n=b ¼ 1:0).
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sffiffiffiffiffiffiffiffiffiffiffiffiffi 4G0 K20 ¼ ðb11 < 0Þ: pb11 Figs. 2 and 3 refer to permeable crack faces. They show, respectively, the variation of the normal stress rzz along the fixed end of the semi-infinite strip for various relative strip widths and locations of the applied concentrated loading. The results in Fig. 2 are obtained for the position of the concentrated force Fz defined
Fig. 3. The variation of the normal stress along the fixed end for various locations of the applied concentrated loading (h=b ¼ 4:0).
Fig. 4. Behaviour of the shear stress along the fixed end under different electric boundary conditions of the crack.
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by n=b ¼ 1:0. Fig. 3 is obtained for the relative strip width defined by h=b ¼ 4:0. In Fig. 2 can be seen that the decrease of the strip width leads to an increase of rzz ðx; 0Þ at all points in the interval ½b; h and especially at the crack tip. The variation of the position of the concentrated force Fz in Fig. 3 can retard or enhance the stress level in the vicinity of the crack tip. Different from what is expected at first sight, closer analysis of the numerical results did not indicate that the curves do intersect in one single point. Fig. 4 shows the influence of the electric crack permeability on rxz ; ðx; 0Þ. The behaviour rxz ðx; 0Þ in the vicinity of the crack tip determines the SIF K2 . The influence of different electric permeabilities is considered: the permeable crack, the limited permeable crack with air as crack medium (er ¼ 1), and the impermeable crack. It is interesting to note that the results for all three cases do not differ essentially. Moreover in the near-crack tip zone the values of stresses for permeable and limited permeable cracks are almost the same and graphically they coincide completely. Figs. 5 and 6 show the effect of the electric permeability of the crack on the intensity factor K20 and on the energy release rate G0 . In the calculation presented here, er varies between 0 (impermeable crack) and 7.
Fig. 5. Dependence of the stress intensity factor K20 on the relative permittivity er of the crack medium for real electric permeability (graph a) and for fictitious medium (graph b).
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Fig. 6. Dependence of the energy release rate G0 on the relative permittivity er of the crack medium for real electric permeability (graph a) and for fictitious medium (graph b).
It is worth a reminder that in reality only crack media with er P 1 exist. It is seen from Figs. 5b and 6b that only cracks with a fictitious crack medium have a more pronounced effect on K20 and G0 due to electric permeability, but for a real electric permeability this effect is very small (Figs. 5a and 6a). It means that rather appropriate values of K20 and G0 can be obtained for any er P 1 by consideration of a completely permeable crack.
7. Conclusion The plane strain state of an interface crack between a rigid conductor and a piezoelectric semi-infinite strip, loaded by a concentrated force and a concentrated electric charge, is considered. Different conditions on the crack faces––permeable, impermeable and limited permeable––are considered. The boundary
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integral relations for a semi-infinite piezoelectric strip are found, and using these relations the system of singular integral equations corresponding to all three types of boundary conditions is formulated. The quasi-invariance of the energy release rate with respect to the artificial contact zone length is confirmed, and the appropriate method of fracture mechanical parameters determination which can be used for a finite sized domain, is suggested. It is found that a electromechanical field can retard or enhance the stress level in the vicinity of the crack tip, depending on the magnitude and the location of the applied concentrated loading. Therefore, optimisation of the magnitude and the location of the applied loading for the sake of decreasing the SIF at the interface crack tip can be suggested. The influence of the electrical permeability of the crack on the electromechanical fields at the crack tip and on the fracture mechanics parameters is considered. It is shown that fracture mechanics parameters of limited permeable cracks differ strongly from the associated values for the impermeable crack, but the results for permeable and limited permeable cracks are almost the same. Therefore, the often used approximation of air-filled impermeable cracks is less suitable then another actively used approximation based upon the completely permeable crack assumption.
Acknowledgements The first author (VG) gratefully acknowledges the support given to him by the Alexander von Humboldt Foundation for carrying out this work at Forschungszentrum Karlsruhe (Institute for Materials Research II).
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