2.1 Basic Equations
For an infinite, homogeneous piezoelectric material, elastic behaviors and electric behaviors are coupled. In other words, applied mechanical loadings cause not only elastic deformation but also electric field. Conversely, applied electric fields produce not only electric displace- ment but also elastic deformation. In general, the former is referred to as piezoelectricity, and the latter is converse piezoelectric effect. Mathematically, in the framework of the theory of linear piezoelectricity, the constitutive equations have four different forms, and usually in fracture mechanics a system of typical constitutive equations take the following form (Ikeda, 1990)
σ =CEs−etE, (1a)
D=es+εsE, (1b)
whereσ, s, D, andE are the stress tensor, strain tensor, electric displacement vector, and electric field vector, respectively; CE, e, and εs are the elastic stiffness tensor measured under a constant electric field condition, the piezoelectric constant tensor, and the dielec- tric permittivity tensor measured a uniform strain condition, respectively. Here et is the transposed matrix ofe.
For a completely anisotropic piezoelectric solid, there are 45 material constants, includ- ing 21 inCE,18 ine,and 6 inεs. However, for a class of common piezoelectric ceramics with point groups6mmexhibiting a transversely isotropic property, the number of involved material constants reduces to 10, where 5 inCE,3 in e,and 2 inεs. For common com- mercially available piezoelectric ceramics such as PZT-4, PZT-5H, etc, relevant material properties are given in Table 1. If we denote the poling direction as x3-axis (or z-axis), and the isotropic plane as the x1x2-plane (or xy-plane), the constitutive equations for a
transversely isotropic piezoelectric ceramic take the following form
σxx =c11sxx+c12syy+c13szz−e31Ez, (2a) σyy =c12sxx+c11syy+c13szz−e31Ez, (2b) σzz =c13sxx+c13syy+c33szz−e33Ez, (2c)
σyz = 2c44syz−e15Ey, (2d)
σxz = 2c44sxz−e15Ex, (2e)
σxy = (c11−c12)sxy, (2f)
Dx= 2e15sxz+ε11Ex, (3a)
Dy = 2e15syz+ε11Ey, (3b)
Dz=e31sxx+e31syy+e33szz+ε33Ez. (3c) Here the components of strain and electric field can be expressed in terms of elastic displacements, and electric potential, by the following equations, respectively,
sij = 1 2
∂ui
∂xj +∂uj
∂xi
, (4a)
Ei =−∂φ
∂xi, (4b)
in whichi, jstand forx, yandz.
In the absence of body forces and free charges, the equilibrium equations of stresses and electric displacements require
∂σxx
∂x +∂σxy
∂y +∂σxz
∂z = 0, (5a)
∂σxy
∂x +∂σyy
∂y +∂σyz
∂z = 0, (5b)
∂σxz
∂x +∂σyz
∂y +∂σzz
∂z = 0, (5c)
and ∂Dx
∂x +∂Dy
∂y +∂Dz
∂z = 0. (6)
Table 1. The relevant material properties
Elastic
stiffnesses (GPa)
Piezoelectric constants (C/m2)
Dielectric permittivities (nF/m) c11 c33 c44 c12 c13 e31 e33 e15 ε11 ε33 PZT-4 139 113 25.6 77.8 74.3 -6.98 13.84 13.44 6.0 5.47
PZT-5H 126 117 35.3 55 53 -6.5 23.3 17 15.1 13
PZT-7 130 119 25 83 83 -10.3 14.7 13.5 17.1 18.6
BaTiO3 150 146 44 66 66 -4.35 17.5 11.4 12.8 15
Figure 1: Cracked piezoelectric material subjected to applied loading.
2.2 Boundary Conditions
To solve an electroelasticity problem of a cracked piezoelectric material, besides the above system of basic governing partial differential equations, appropriate boundary conditions must be furnished. In general, at the boundary of a piezoelectric material, two kinds of boundary conditions may be applied, one arising from mechanical part and the other arising from electric part; so it is assumed (Fig. 1) that
σijnj|Γm =Ti, (x, y, z)∈Γm, (7a) ui|Γm =Ui, (x, y, z)∈Γcm, (7b) and
Djnj|Γe =D0, (x, y, z)∈Γe, (8a) φ|Γe =φ0, (x, y, z)∈Γce, (8b) where Γm,Γcm, Γe, and Γce denote partial boundaries, Γm ∪Γcm = Γe∪ Γce is the total boundary,n= (n1, n2, n3)andt= (t1, t2, t3)are respectively the directions of the outward normal and tangential vectors of the boundary surface involved. HereTi, Ui are prescribed stress and displacement, andD0, φ0are prescribed charge and potential, respectively.
For a crack problem, the boundary conditions at the crack surfaces are of significance.
As usual, the mechanical boundary conditions at the crack surfaces are clearly free of trac- tion, which can be written as
σn= 0, σt= 0, (x, y, z)∈Σ, (9) where Σstands for the crack surfaces. For the electric boundary conditions at the crack surfaces, the situation becomes more complicated since the electric fields are permeable relative to an opening crack, i.e.
D+n = ¯Dn−, φ+= ¯φ−, (x, y, z)∈Σ, (10a) D−n = ¯Dn+, φ−= ¯φ+, (x, y, z)∈Σ, (10b)
where a quantity with a bar denotes the one in the crack interior.
In fact, due to the opening of crack, the crack interior such as vacuum can be treated as a dielectric; so there are an electric potential difference across the opening crack. In other words,φ¯+ = ¯φ−.Such a statement has been supported by an experimental evidence in Schneider et al. (2003), who observed a distinct drop of electric potential between two surfaces of an opening crack, which also implies the existence of electric field in the opening dielectric crack interior. Furthermore, as compared to the crack length, the opening height is very small, which allows us to assumeE¯nin the crack interior to be a constant, given by
E¯n=−∆φ
∆u, (11)
where ∆u and∆φare the jumps of elastic displacement and potential across the crack.
Hence, the electric displacement inside the opening crack is governed by the following relation
D¯n=−¯ε∆φ
∆u. (12)
This is equivalent to that given by Hao and Shen (1994), Wang and Jiang (2002). Note that ∆u and ∆φ are last values of crack opening displacement and potential difference across the crack surfaces posterior to deformation, not prior to deformation. If enforcing that the above relation holds before deformation, it meansD+n =D−n, φ+=φ−, equivalent to the assumption of the so-called permeable crack. Consequently, due to the opening of a crack, the results corresponding to the permeable crack certainly exist some errors as compared to those for a realistic crack. Therefore, the above relation (12) in fact reflects a nonlinear procedure where initial electric boundary conditions are dominated by last values of∆uand∆φ. It is further noted that the selection of such boundary conditions differ from formulation of elastic boundary conditions since elastic boundary conditions are described based on the boundary surface prior to deformation, rather than posterior to deformation.
On the other hand, another kind of electric boundary conditions has been formulated by Gao and Fan (1999), and Zhang et al. (2002), who treated a crack as a limiting case of an elliptical hole by setting a shorter axis to approach zero. Clearly, for a real elliptical hole such a treatment gives an exact solution. In other words, electric boundary conditions are also described based on the boundary before deformation. However, for a crack having no thickness prior to deformation, the crack surfaces are electrically contacting. Under the action of applied loadings, the crack may either open or close depending on the magni- tude and direction of mechanical and electric loadings. Therefore, it is believed that (12) is superior to other electric boundary conditions. Moreover, two usual impermeable and conducting cracks can be taken as two limiting cases. The former corresponds toε¯→ 0, which yieldsD¯n= 0,in agreement with the so-called impermeable electric boundary con- dition (Pak, 1990). This case imposes the dielectric permittivity of crack interior to vanish, and the electric displacement in the crack interior and at the boundary are equal to zero.
The latter corresponds toε¯→ ∞,which leads toφ+ =φ− = 0,coinciding with the so- called conducting electric boundary conditions. Obviously, for a finite value ranging from 0 to∞,the desired results must lie between those of two limiting cases. It is interesting to point out that for certain cases, the conducting crack is equivalent to a permeable crack, i.e. adopting (12) before deformation meaning∆φ= 0, which specifies continuous electric displacement and potential across the crack surfaces (Parton, 1976).