A mathematical strip-saturation model for a piezoelectric plane weakened by two collinear equal cracks

1. Introduction

Piezoelectric ceramics have proved their wide utility as sensors, actuator, or transducers in hi-tech instruments. Because of their brittle characteristics the fracture study of such materials has drawn the attention of researchers. A lot of crack problems have been investigated since the 1990s. More recently multisite crack problems have also been addressed.

Multi-cack damage problems being more often encountered in real life situation have attracted the attention of various research communities. While a lot of work has been done on single crack problems in piezoelectric media [1–4], comparatively less work is reported for two or more cracks weakening a piezoelectric plate/strip. Examples are as follows: the generalized 2-D problems in piezoelectric media with collinear cracks based on Stroh formalism and the exact electric boundary conditions on the crack faces [5]; the problem of two equal collinear cracks situated normal to the strip boundaries in an infinitely long piezoelectric strip addressed by using the Fourier series method [6]; a closed-form solution for the anti-plane mechanical and the in-plane electric and magnetic field for two collinear cracks in a magneto-electro-elastic layer of finite thickness under the condition of permeable crack faces using the integral transform method [7]; the problem of two collinear unequal cracks in a piezoelectric material subjected to mode-I loading by a new approach of real fundamental solutions [8].

The strip-yield models for piezoelectric ceramics [9] was started by considering a strip electric saturation model for a finite crack perpendicular or parallel to the poling axis weakening an infinite poled piezoelectric ceramics medium with electric polarization reaching a saturation limit along a line segment in front of the crack. While Wang [3] gave a fully anisotropic analysis of the strip electric saturation model proposed in Gao et al. [9].

Only a small amount of work has been reported for mechanical and electric yielding of a cracked piezoelectric media. The first one to propose a strip-electric saturation and mechanical yield model for mode-III interface crack for piezoelectric material was Shen et al. [10]. Their work was further extended by Bhargava et al. [11] for a cracked piezoelectric plate. An electrical and mechanical yield zone problem for a crack in a thin ductile layer between two piezoelectric materials under remote electromechanical loading was given in Loboda et al. [12]. More recently, we [13] have given a strip-saturation model for a piezoelectric plane weakened by two collinear cracks with coalesced interior zones.

The multiple-crack problems for piezoelectric ceramics considered until now have not yet addressed the crack opening arrest problem. The present work addresses this paucity. In section 2, the fundamental formulation and solution methodology are presented, which is recapitulated from Gao et al. [14]. Section 4 illustrates the solution of the physical problem stated in section 3, for a poled transversely isotropic piezoelectric plane weakened by two collinear equal straight cracks. In section 5, the analytical closed-form expressions are derived for fracture parameters such as stress intensity factor (SIF), saturation zone length, crack opening displacement (COD), crack opening potential drop (COP), and energy release rate (ERR). A qualitative numerical case study is presented in section 6 for ceramics PZT-4, PZT-5H, and BaTiO₃ to study the effects of various fracture parameters such as developed saturation zone length and prescribed load, SIF, ERR, COD, and COP on crack growth resistance. The ERR and SIF variations are investigated with respect to the inter-crack distance.

2. Fundamental formulation and solution methodology

As are well-known, in a rectangular coordinate system, x_i(i = 1, 2, 3), the basic equations for stress components, σ_ij, and electric displacement component, D_i, (i,j = 1, 2, 3), may be expressed as follows: the constitutive equations are

σ i j = c i j k l γ k l − e k i j E k , D k = e k i j γ i j + ε k l E l ,

(1)

tradient equations are

γ i j = 1 2 ( u i , j + u j . i ) , E i = − φ , i ,

(2)

and the equilibrium equations for stresses and electric displacement in the absence of body force and electric charge, respectively, can be written as

σ i j , j = 0 , D i , i = 0 ,

(3)

where u_i, φ, γ_ij, and E_i denotes displacement, the electric potential, the strain, and the electric field, respectively. c_ijkl, e_ijk, and ε_ij stand for the elastic constants, the piezoelectric constants, and the dielectric constants, respectively.

The methodology presented here is recapitulated from Gao et al. [14] to make the paper self-sufficient for the reader.

For a two-dimensional problem all the field variables depend on x₁ and x₂ and are independent of x₃. Therefore, we introduce a generalized displacement vector, u, from Barnett et al. [15] as

u = [ u 1 , u 2 , u 3 , φ ] T = a f ( x 1 + p x 2 ) ,

(4)

where superscript, ‘T’, denotes the transpose of the matrix, f (x₁ + px₂), is an analytic function, p is a complex number, and a is a constant four element column vector. Equations (1)–(3) satisfy equation (4) for an arbitrary f (x₁ + px₂) if

[ W + p ( R + R T ) + p 2 Q ] a = 0 ,

(5)

which has a non-trivial solution only if

| W + p ( R + R T ) + p 2 Q | = 0 .

(6)

Here the matrices W, R, and Q are given by

W = [ c i 1 k 1 e 11 i e 11 i T − ε 11 ] , R = [ c i 1 k 2 e 21 i e 12 i T − ε 12 ] , Q = [ c i 2 k 2 e 22 i e 22 i T − ε 22 ] , i , k = 1 , 2 , 3 .

(7)

Let the eight roots of equation (6) be denoted by p_α and p ¯ α ( α = 1 , 2 , 3 , 4 ) . The conditions of positive definiteness of the strain energy and electrical energy densities give that p_α is not real. To determine p_α the following standard eigen-equations are solved

N ζ = p ζ ,

where

N = ( N 1 N 2 N 3 N 1 T ) , ζ = ( a b ) , N 1 = − Q − 1 R T , N 2 = Q − 1 = N 2 T , N 3 = R Q − 1 R T − W = N 3 T .

According to the Stroh formulation the general solution satisfying equations (1)–(3) may be written as

u , 1 = A F ( z ) + A ¯ F ( z ) ¯ ,

(8)

Φ , 1 = B F ( z ) + B ¯ F ( z ) ¯ ,

(9)

where

A = ( a 1 , a 2 , a 3 , a 4 ) , B = ( b 1 , b 2 , b 3 , b 4 ) , F ( z ) = d f ( z ) d z , f ( z α ) = [ f 1 ( z 1 ) , f 2 ( z 2 ) , f 3 ( z 3 ) , f 4 ( z 4 ) ] T , z α = x 1 + p α x 2 .

The column vector of matrix B = (b₁, b₂, b₃, b₄) is related to the column vector of matrix A = (a₁, a₂, a₃, a₄) in the following form

b k = ( R T + p k Q ) a k , k = 1 , 2 , 3 , 4 ,

and Φ is the generalized stress function such that

σ 2 = [ σ 2 j , D 2 ] T = Φ , 1 , σ 1 = [ σ 1 j , D 1 ] T = − Φ , 2 .

(10)

According to Muskhelishvili [16], the stress component, σ₂₂, can be expressed in terms of two complex potentials, Ψ(z) and χ(z), as

σ 22 = Ψ ( z ) + χ ( z ¯ ) − ( z − z ¯ ) Ψ ′ ( z ) ¯ ,

(11)

Ψ + ( t ) + χ − ( t ) = σ 22 + , Ψ − ( t ) + χ + ( t ) = σ 22 − .

(12)

Consider a plate occupying the x₁ox₂ plane and cut along n-collinear straight cracks L_i(i = 1,2,….n), with end points a_i, b_i, lying on ox₁–axis. The rims of the cracks are subjected to uniform constant stress, σ₂₂; then, equation (12) yields, following Hilbert problems under the assumption lim x 2 → 0 x 2 Ψ ′ ( t + i x 2 ) = 0 , where t being any point on L_i(i = 1,2,….n)

[ Ψ ( t ) + χ ( t ) ] + + [ Ψ ( t ) + χ ( t ) ] − = σ 22 ,

(13)

[ Ψ ( t ) − χ ( t ) ] + − [ Ψ ( t ) − χ ( t ) ] − = 0 .

(14)

The superscripts + and − denote the value of the function as it is approach from x₂ > 0 or x₂ < 0, respectively. The solution of which yields

Ψ ( z ) + χ ( z ) = 1 2 π i X ( z ) ∫ L X ( t ) σ 22 t − z d t + 2 P n ( z ) X ( z ) ,

(15)

Ψ ( z ) − χ ( z ) = − Γ ¯ ′ ,

(16)

where

X ( z ) = ∏ k = 1 n ( z − a k ) 1 / 2 ( z − b k ) 1 / 2 , P n ( z ) = C o z n + C 1 z n − 1 + … . + C n , Γ ¯ ′ = − 1 2 ( N 1 − N 2 ) e − 2 β i ,

and N₁, N₂ being the values of principle stresses prescribed at infinity, β is the angle between N₁ and ox₁-axis. The constant C_o is determined using a boundary condition at infinity, and constants C_i(i = 1, 2, ….n) are determined using the condition of single-valuedness of displacements on crack rims.

3. Statement of the problem

A transversely isotropic piezoelectric plane occupying an entire x₁ox₂ plane, poled along the positive ox₂-direction is considered. The plane is cut along the two equal collinear hairline straight cracks, L₁ and L₂. The cracks L₁ and L₂ occupy the respective intervals [d, c] and [−c, −d] on the x₁-axis. Uniform constant stress, σ 22 = σ 22 ∞ , and electrical displacement, D 2 = D 2 ∞ , are prescribed at the remote boundary of the plane; consequently cracks open in a similar fashion forming a strip-saturation zone ahead of each tip of the cracks. The respective saturation zones Γ₁, Γ₂, Γ₃, and Γ₄ developed at each tip, c, d, −d, and −c, occupy the respective intervals [c, a], [b, d],[−d, −b], and [−a, −c] on the x₁-axis. The developed saturation zones are arrested by a normal cohesive saturation limit electric displacement, D_y = D_s. The schematic configuration of the problem is depicted in Figure 1.

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Figure 1.

Schematic representation of the problem.

The boundary conditions of the problem may be mathematically expressed as

4. Solution of the problem

Boundary condition (i) and equation (9) lead to following Hilbert problem

[ B F ( x 1 ) − B ¯ F ¯ ( x 1 ) ] + − [ B F ( x 1 ) − B ¯ F ¯ ( x 1 ) ] − = 0 .

(17)

Consequently its solution is written using equation (16) as

B F ( z ) − B ¯ F ¯ ( z ) = 0 .

(18)

Using boundary condition (iv) and equation (9), one obtains

h + ( x 1 ) + h − ( x 1 ) = − V , d < | x 1 | < c

(19)

where h ( z ) = B F ( z ) = B ¯ F ¯ ( z ) .

This introduces

Ω ( z ) = H R B F ( z ) , h ( z ) = Λ Ω ( z ) ,

(20)

where Λ = [H^R]⁻¹, H^R = 2ReY, Y = Im(AB⁻¹), i,j = 1, 2, 3.

One then writes equation (19) in component form as

Λ 22 [ Ω 2 + ( x 1 ) + Ω 2 − ( x 1 ) ] + Λ 24 [ Ω 4 + ( x 1 ) + Ω 4 − ( x 1 ) ] = − σ 22 ∞ , d < | x 1 | < c

(21)

Λ 42 [ Ω 2 + ( x 1 ) + Ω 2 − ( x 1 ) ] + Λ 44 [ Ω 4 + ( x 1 ) + Ω 4 − ( x 1 ) ] = − D 2 ∞ , d < | x 1 | < c .

(22)

Eliminating Ω 4 + ( x 1 ) + Ω 4 − ( x 1 ) from equations (21) and (22), one obtains

Ω 2 + ( x 1 ) + Ω 2 − ( x 1 ) = − ( σ 22 ∞ Λ 44 − D 2 ∞ Λ 24 ) / ( Λ 22 Λ 44 − Λ 24 Λ 42 ) , d < | x 1 | < c

(23)

The general solution of equation (23) using equation (15) may be written as

Ω 2 ( z ) = 1 2 σ 22 ∞ Λ 44 − D 2 ∞ Λ 24 Λ 22 Λ 44 − Λ 24 Λ 42 { z 2 − c 2 λ 2 X ( z ) − 1 } ,

(24)

where k 2 = ( c 2 − d 2 ) / c 2 , λ 2 = E ( k ) / F ( k ) , X ( z ) = ( z 2 − d 2 ) ( z 2 − c 2 ) .

Analogously to determining Ω₄(z), equation (22) is solved using boundary condition (ii) and equation (15), the solution may be written as

Ω 4 ( z ) = D s π Λ 44 ( π 2 − θ d + θ c ) − D s π Λ 44 { ( z 2 − a 2 λ 1 2 ) ( π 2 − ψ d + ψ c ) + a 2 R 1 − π 2 D 2 ∞ D s ( z 2 − a 2 λ 1 2 ) } 1 X 1 ( z ) − D 2 ∞ / 2 Λ 44 − Ω 2 ( z ) Λ 42 / Λ 44 ,

(25)

where

X 1 ( z ) = ( z 2 − a 2 ) ( z 2 − b 2 ) , k 1 2 = ( a 2 − b 2 ) / a 2 , λ 1 2 = E ( k 1 ) / F ( k 1 ) , s i n 2 ψ d = ( a 2 − d 2 ) / ( a 2 − b 2 ) , sin 2 ψ c = ( a 2 − c 2 ) / ( a 2 − b 2 ) ,

and θ_d, θ_c, R₁ can be found in appendix A.

5. Applications

Closed-form expressions are derived for SIF, saturation zone lengths, COD, COP, and ERR.

5.1. Stress intensity factor

The SIF, K_I (at the crack tip), is determined using the definition at the tip x₁ = d

K I ( d ) = lim x 1 → d − 2 π ( d − x 1 ) σ 22 ( x 1 ) ,

(26)

and at the tip x₁ = c

K I ( c ) = lim x 1 → c + 2 π ( x 1 − c ) σ 22 ( x 1 ) .

(27)

Evaluating σ₂₂(x₁) using equations (9), (20), and (24) and boundary condition (iii) and substituting the result into equations (26) and (27) and simplifying, one obtains at the tip x₁ = d

K I ( d ) = − ( σ 22 ∞ − D 2 ∞ Λ 24 / Λ 44 ) π d ( c 2 − d 2 ) ( d 2 − c 2 λ 2 ) ,

(28)

and at the tip x₁ = c

K I ( c ) = ( σ 22 ∞ − D 2 ∞ Λ 24 / Λ 44 ) π c ( c 2 − d 2 ) ( c 2 − c 2 λ 2 ) .

(29)

5.2. Saturation zone size

The electric displacement ahead of the crack tip is determined using

D 2 ( x 1 ) = Λ 42 [ Ω 2 + ( x 1 ) + Ω 2 − ( x 1 ) ] + Λ 44 [ Ω 4 + ( x 1 ) + Ω 4 − ( x 1 ) ]

(30)

Substituting the values of Ω₂(x₁) and Ω₄(x₁) from equations (24) and (25) and simplying we obtain

D 2 ( x 1 ) = 2 D s π ( π 2 − θ d + θ c ) + D 2 ∞ { x 1 2 − a 2 λ 1 2 X 1 ( x 1 ) − 1 } − 2 D s π { ( x 1 2 − a 2 λ 1 2 ) ( π 2 − ψ d + ψ c ) + a 2 R 1 } 1 X 1 ( x 1 ) .

(31)

Extending the Dugdale hypothesis for electric displacement to remain finite at every point of a piezoelectric ceramic under the linear piezoelectricity theory assumption, one obtains two non-linear equations, one at the tip x₁ = b

( b 2 a 2 − λ 1 2 ) ( π D 2 ∞ 2 D s − π 2 + ψ d − ψ c ) − R 1 = 0 ,

(32)

and one at the other tip x₁ = a

( 1 − λ 1 2 ) ( π D 2 ∞ 2 D s − π 2 + ψ d − ψ c ) − R 1 = 0 .

(33)

These results enable one to determine a and b, and the saturation zone is than determined by (a − c) and (d − b), respectively.

5.3. Crack opening displacement

The COD, giving the relative opening of the faces of the crack, is defined as the jump displacement vector as

i Δ u , 1 = i [ u 1 , 1 + − u 1 , 1 − , u 2 , 1 + − u 2 , 1 − , u 3 , 1 + − u 3 , 1 − , E 1 − − E 1 + ] T

(34)

Using equations (8) and (24), one obtains

Δ u 2 , 1 = Ω 2 + ( x 1 ) − Ω 2 − ( x 1 ) ,

Δ u 2 ( x 1 ) = − 2 c ( σ 22 ∞ Λ 44 − D 2 ∞ Λ 24 ) Λ 22 Λ 44 − Λ 24 Λ 42 { λ 2 F ( ϕ , k ) − E ( ϕ , k ) } ,

(35)

where sin 2 ϕ = ( c 2 − x 1 2 ) / ( c 2 − d 2 ) .

5.4. Crack opening potential drop

The COP is obtained equating the fourth component of equation (34) and equation (25) as at the tip x₁ = d

Δ u 4 ( d ) = − D s π Λ 44 { R 3 − π a D 2 ∞ D s R 4 } ,

(36)

and at the tip x₁ = c

Δ u 4 ( c ) = D s π Λ 44 { R 5 − π a D 2 ∞ D s ( E ( ψ c , k 1 ) − λ 1 2 F ( ψ c , k 1 ) ) } ,

(37)

where R₃, R₄, and R₅ can be found from appendix A.

5.5. Energy release rate

The local ERR is calculated [17] using the equation at the tip x₁ = d

J ( d ) = π 2 d ( c 2 − d 2 ) ( d 2 − c 2 λ 2 ) 2 [ V T H R V − 1 Λ 44 ( D 2 ∞ ) 2 ] = π 2 d ( c 2 − d 2 ) ( d 2 − c 2 λ 2 ) 2 { H ( 2 , 2 ) ( σ 22 ∞ ) 2 + 2 H ( 2 , 4 ) σ 22 ∞ D 2 ∞ + ( H ( 2 , 4 ) ) 2 H ( 2 , 2 ) ( D 2 ∞ ) 2 } ,

(38)

and at the tip x₁ = c

J ( c ) = π 2 c ( c 2 − d 2 ) ( c 2 − c 2 λ 2 ) 2 [ V T H R V − 1 Λ 44 ( D 2 ∞ ) 2 ] = π 2 c ( c 2 − d 2 ) ( c 2 − c 2 λ 2 ) 2 { H ( 2 , 2 ) ( σ 22 ∞ ) 2 + 2 H ( 2 , 4 ) σ 22 ∞ D 2 ∞ + ( H ( 2 , 4 ) ) 2 H ( 2 , 2 ) ( D 2 ∞ ) 2 } .

(39)

The apparent ERR or global energy release rate, J_a, at the inner and outer crack tips is calculated using

J a ( d ) = J ( d ) + D s Δ u 4 ( d ) ,

(40)

J a ( c ) = J ( c ) + D s Δ u 4 ( c ) .

(41)

6. Case study

A case study is presented to investigate the behaviour of various parameters, such as saturation zone size, SIF, COD, ERR, as the applied load is increased, resisting the crack growth. The material constants are given in Table 1, which are taken from Ou et al. [18]. We assumed that the lengths of the cracks are 10 mm and the saturation limit electric displacement, D_s = 0.03C/m², which is taken from Loboda et al. [19].

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Table 1.

The material properties constants.

Material	c 11	c 12	c 13	c 33	c 44	e 13	e 33	e 15	ε 11	ε 33
PZT-4	139	77.8	74.3	113	25.6	–6.98	13.8	13.4	6.0	5.47
PZT-5H	126	55	53	117	35.3	–6.50	23.3	17	15.1	13
BaTiO3	150	66	66	146	44	–4.35	17.5	11.4	9.87	11.2

Units: elastic constant, 10⁹ N.m⁻²; piezoelectric constant, C.m⁻²; dielectric constant, 10⁻⁹ C (V.m)⁻¹.

Variation of SIF, K_I, versus the normalized inter-crack distance is plotted in Figure 2 for different piezoelectric ceramics and D 2 ∞ = 0 . 005 C / m 2 , σ 22 ∞ = 10 M P a . It is observed that the SIF at the inner crack tip is higher than that at the exterior tip. It is to be noted that when the two cracks are nearer to each other there is a higher stress singularity at the interior tip; this is because of the mutual interactions of the two cracks on each other and because the stress singularity decreases, stabilizes, and coincides with the results of the single centre crack problem as the inter-crack distance increases. It is also observed that the SIF is maximized for ceramic PZT-4 when compared to the other ceramics, PZT-5H and BaTiO₃. This is because of the dependence of K_I on the components of Irwin’s material constant matrix, H^R, and the ratio Λ₂₄/Λ₄₄. It is maximized for ceramic PZT-4 and minimized for PZT-5H. The numerical values of the components of Irwin’s matrix, H^R, are given in Table 2 for all the three ceramics.

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Figure 2.

SIF versus inter-crack distance for different piezoelectric ceramics.

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Table 2.

Components of Irwin’s matrix H^R and its inverse matrix Λ = [H^R]⁻¹ for different piezoelectric ceramics.

Material	H22	H24	H44	Λ24	Λ44
PZT-4	3.4989 × 10−11	0.0443	−1.7486 × 108	5.4817	−4.3303 × 10−9
PZT-5H	3.2131 × 10−11	0.0256	−0.9156 × 108	7.1078	−8.9384 × 10−9
BaTiO3	2.8303 × 10−11	0.0244	−1.4642 × 108	5.1534	−5.9695 × 10−9

Units: H(2, 2), N⁻¹ m²; H(2,4), C⁻¹ m²; H(4,4), C⁻¹ (V.m); Λ₂₄, N(V.m)⁻¹; Λ₄₄, C(V.m)⁻¹.

SIF, K_I, is plotted against increasing prescribed electric displacement in Figure 3 for different piezoelectric ceramics. It is observed that as the load is increased the SIF also increases both at the inner and outer tips of the crack. It is to be noted that the SIF is considerable higher at the inner tip vis-a-vis that at the exterior tip of the crack for all the ceramics considered. Ceramic PZT-4 shows the highest stress concentration and ceramic PZT-5H shows the least concentration.

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Figure 3.

SIF versus electric displacement load for different piezoelectric ceramics.

Figure 4 depicts the saturation zone sizes for ceramic PZT-4 at the interior and exterior tips of the crack when the electric load ratio, D 2 ∞ / D s , is increased. It is observed that the zone developed at the interior tip is bigger than that at the exterior tip by the same applied load, D 2 ∞ / D s . And as the prescribed load is increased then the developed zone size also increases at both the interior and exterior tips, as expected. Also the zone size is more when the two cracks are situated close to each other.

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Figure 4.

Normalized saturation zone length versus electric displacement load ratio for PZT-4 ceramic.

COD over the rims of the cracks is plotted in Figure 5. The dotted line shows the COD for a single centre crack problem for ceramic BaTiO₃. It may be noted that the crack rims open symmetrically with respect to the middle point of the crack. The COD for two symmetrically situated cracks shows a shift in COD toward the internal crack tip. Also, the COD is larger in this case as compared to the single centre crack, as expected. It may also be noted from Figure 5 that ceramic PZT-4 has the maximum opening and that the crack opens less for ceramic BaTiO₃.

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Figure 5.

COD profile over the crack surface.

Variation of COP versus normalized inner and outer saturation zones is drawn in Figure 6(a) and (b) for ceramic PZT-4. It is observed that the potential drop is more at the inner zone than that at outer zone, as expected. As the inter-crack distance, 2d/(c − d), is increases, the COP decreases at both the crack tips, but a drop in COP is higher at the inner tip when compared with that at the outer tip.

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Figure 6.

COP at the (a) inner and (b) outer saturation zones for PZT-4 ceramic.

Figure 7 depicts the variation of the local ERR at the inner and outer crack tips versus the normalized inter-crack distance for different piezoelectric ceramics. It can be seen from the Figure 7 that the local ERR is more at the inner crack tip than the outer crack tip. It is also observed that with an increase in inter-crack distance the local ERR at the inner and the outer tips becomes equal. This is because of the mutual influence of the cracks on each other decreases as the distance between them increased. Also, this is because the local ERR depends on factor V^THV. It is maximized for ceramic PZT-4 and minimized for ceramic BaTiO₃; therefore, the local ERR is maximized for ceramic PZT-4 and minimized for ceramic BaTiO₃.

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Figure 7.

Local ERR versus inter-crack distance for different piezoelectric ceramics.

Figure 8 depicts the variation in the global ERR versus the applied mechanical loading, σ 22 ∞ , for ceramic PZT-4 and different inter-crack distances. It is found that the increase in mechanical loading always increases the global ERR. This is true at both the inner and outer tips of the cracks. The mutual influence of two cracks on each other is clearly seen as the global ERR is more at the inner tip than at the outer tip.

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Figure 8.

Global ERR versus applied mechanical loading for different inter-crack distance and PZT-4 ceramic.

Figure 9 shows the variation of the normalized global ERR versus the applied electric loading, D 2 ∞ , for ceramic PZT-4 and different inter-crack distances. It shows that the normalized global ERR decreases as the applied electric displacement loading increases. Again, as in Figure 8, the mutual influence of two cracks on each other is clearly seen.

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Figure 9.

Normalized global ERR versus applied electric displacement load for different inter-crack distance and PZT-4 ceramic.

7. Conclusion