Dielectric Crack Problems

4 Three-Dimensional Problems

4.2 Dielectric Crack Problems

For simplicity, consider a piezoelectric material with a penny-shaped crack of radius a occupying|r|< a,0≤θ≤2π, z= 0, as shown in Fig. 10.

For determination of electroelastic field induced by a penny-shaped crack, based on axisymmetry of the problem in question, we employ the Hankel transform to analyze it.

To this end, one can take three generalized harmonic functions by Hankel transform of the zeroth order

F_j(r, z) =− ^∞

0

1

ξA_j(ξ)e^−γjξzJ₀(ξr)dξ, (104)

for z ≥ 0,where Aj(ξ)’s are unknown functions to be determined through appropriate electric and elastic boundary conditions. Making use of superposition of a uniform field of a piezoelectric space without crack subjected to far-field loading (89), an appropriate

solution can be given as follows u_r(r, z) =

3 j=1

∞

0 A_j(ξ)e^−γjξzJ₁(ξr)dξ+B₁r, (105a) u_z(r, z) =

3 j=1

η_3jγ_j

∞

0 A_j(ξ)e^−γjξzJ₀(ξr)dξ+B₃z, (105b) φ(r, z) =

3 j=1

η_4jγ_j

∞

0 A_j(ξ)e^−γjξzJ₀(ξr)dξ+B₄z, (105c) whereB_k(k= 1,3,4)are unknown constants. Moreover, expressions for the stresses and electric displacements in terms ofA_j(ξ)are

σ_rr(r, z) =− 3 j=1

β_0j

∞

0 ξA_j(ξ)e^−γjξzJ₀(ξr)dξ− c₁₁−c₁₂

r ×

3 j=1

∞

0 A_j(ξ)e^−γjξzJ₁(ξr)dξ+ (c₁₁+c₁₂)B₁+c₁₃B₃+e₃₁B₄, (106a) σ_zz(r, z) =−³

j=1

β_1j

∞

0 ξA_j(ξ)e^−γjξzJ₀(ξr)dξ+ 2c₁₃B₁+c₃₃B₃+e₃₃B₄, (106b) σrz(r, z) =−

3 j=1

β2j

∞

0 ξAj(ξ)e^−γjξzJ1(ξr)dξ, (106c)

D_r(r, z) =−³

j=1

β_3j

∞

0 ξA_j(ξ)e^−γjξzJ₁(ξr)dξ, (107a)

D_z(r, z) =−³

j=1

β_4j

∞

0 ξA_j(ξ)e^−γjξzJ₀(ξr)dξ+ 2e₃₁B₁+e₃₃B₃−ε₃₃B₄, (107b) whereβ_kj(k= 0,1, ...,4, j = 1,2,3)are defined as before.

From symmetry of the problem, one clearly gets the shear stress at the crack plane vanishing, namely

σ_rz(r,0) = 0, 0≤r <∞. (108) Since attention is restricted to the upper half-space, the following condition

u_z(r,0) = 0, φ(r,0) = 0, r ≥a (109) must be supplemented owing to symmetry of the problem. Besides, at the crack surfaces, electromechanical boundary conditions

σ_zz(r,0) = 0, r < a, (110a) D_z(r,0) = ¯D, r < a, (110b)

should be satisfied.

First, from the remote electromechanical loading expressed by (89), we can get a system of linear equations, which can be used to determine uniquelyB_k(k = 1,3,4). The final result is

B₁ = (c₃₃e₃₁−c₁₃e₃₃)E^∞−c₁₃σ^∞

(c₁₁+c₁₂)c₃₃−2c²₁₃ , (111a)

B₃ = [(c11+c12)e33−2c13e31]E^∞+ (c11+c12)σ^∞

(c₁₁+c₁₂)c₃₃−2c²₁₃ , (111b)

B₄ =−E^∞, (111c)

for prescribed stress and electric field at infinity, or B₁ = e₃₁E^∞−c₁₃s^∞

c₁₁+c₁₂ , (112a)

B₃ =s^∞, (112b)

B₄ =−E^∞, (112c)

for prescribed strain and electric field at infinity.

WithB_k(k= 1,3,4)at hand, one can further seek the disturbed electroelastic field of a piezoelectric body weakened by a penny-shaped crack. Utilizing the boundary conditions (108) yields

3 j=1

β_2jA_j(ξ) = 0. (113)

On the other hand, from (110) one has

−³

j=1

β_1j

∞

0 ξA_j(ξ)J₀(ξr)dξ+σ₀= 0, r < a, (114a)

− 3 j=1

β_4j

∞

0 ξA_j(ξ)J₀(ξr)dξ+D₀= ¯D, r < a, (114b) with

D¯ 3 j=1

η_3jγ_j

∞

0 A_j(ξ)J₀(ξr)dξ=−¯ε 3 j=1

η_4jγ_j

∞

0 A_j(ξ)J₀(ξr)dξ, r < a, (115) where

σ₀=σ^∞, (116a)

D₀= (c₁₁+c₁₂)e₃₃−2c₁₃e₃₁ (c11+c12)c33−2c²₁₃ σ^∞ +

(c₁₁+c₁₂)e²₃₃+ 2c₃₃e²₃₁−4c₁₃e₃₃e₃₁ (c11+c12)c33−2c²₁₃ +ε₃₃

E^∞, (116b)

for prescribed stress and electric field at infinity, or σ₀=

c₃₃− 2c²₁₃ c₁₁+c₁₂

s^∞+

2c₁₃e₃₁ c₁₁+c₁₂ −e₃₃

E^∞, (117a)

D₀=

e₃₃− 2c₁₃e₃₁ c₁₁+c₁₂

s^∞+

ε₃₃− 2e²₃₁ c₁₁+c₁₂

E^∞. (117b)

for prescribed strain and electric field at infinity.

Next by introducing a new intermediate auxiliary functionA(ξ)such that

Aj(ξ) =ajA(ξ) (118)

wherea_j’s are constants, we have

3 j=1

β_2ja_j = 0, (119a)

3 j=1

Dη¯ _3j+ ¯εη_4j

γ_ja_j = 0, (119b)

σ₀ 3 j=1

β4jaj+D¯ −D0³

j=1

β1jaj = 0. (119c)

In order to obtain a non-trivial solution of the above system, the determinant of the coefficient matrix must take zero, from which D¯ can be determined. By comparing the above result with that in the previous section, we find that the desired expression forD¯ is the same as (56) together with (57). So the dependence ofD¯ is identical. It is worth noting that althoughD¯as a function ofσ₀andD₀is the same for two- and three-dimensional cases, the relation betweenD¯ andE^∞is not identical. For example, the variations ofD¯ withE^∞ under different far-field conditions are shown forσ^∞ = 8MPa ors^∞ = 2×10⁻⁴ in Fig.

11. By comparison, one can find that for the two-dimensional case under prescribed stress σ^∞= 8MPa, the critical point forε_r = 0is at aboutE^∞ =−8.2kV/cm, while for three- dimensional case, the critical point is at aboutE^∞ =−7.5kV/cm. Moreover, in the case of prescribed strains^∞ = 2×10⁻⁴,the impermeable crack does not close forE^∞ < 18 kV/cm for the two-dimensional case, whereas the crack closes at aboutE^∞ = 14 kV/cm for the three-dimensional case, which can be found in Fig. 11. On the other hand, when using the permeable assumption or conducting assumption, as well as (12), we observe that the electric displacement is identical forE^∞lying in a range greater than8.4kV/cm for the two-dimensional case, and6.6kV/cm for the three-dimensional case, respectively. It turns out that when a crack closes, the electric displacement at the crack surfaces coincide with those for a permeable crack.

The remaining is to determineA(ξ).From the boundary conditions, we get dual inte- gral equations forA(ξ)as follows

∞

0 ξA(ξ)J₀(ξr)dξ =Q, r < a, (120a)

∞

0 A(ξ)J₀(ξr)dξ = 0, r ≥a, (120b)

Figure 11: Electric displacement at the crack surfaces for a cracked PZT-4 as a function of applied electric field for a three-dimensional case, a)σ^∞= 8MPa, b)s^∞= 2×10⁻⁴.

whereQis still defined as (52).

To solve the above dual integral equations, we chooseA(ξ)expressed in terms of a new functionϕ(t)through the integral

A(ξ) =ξ

a

0 tϕ(t)J₀(ξt)dt. (121)

Eq. (120b) is completely satisfied when inserting (121) into (120b). On the other hand, substituting (121) into (120a) leads to

a

0 tϕ(t)dt

∞

0 ξ²J0(ξt)J0(ξr)dξ =Q, r < a. (122) Now remembering the property of Bessel function (67) and the following result

∞

0 J1(ξt)J1(ξr)dξ= 2 πrt

min(t,r) 0

s²

√r²−s²√

t²−s²ds, (123) we can convert (122) to the following equation

a

0 ϕ(t)dtd dt

min(t,r) 0

s²

√r²−s²√

t²−s²ds= π

4Qr², r < a, (124)

or r

0

s²

√r²−s²ds

a s

ϕ(t)

√t²−t²dt=−π

4Qr², r < a, (125) where in deriving the above equation, we have utilized the property

a 0 dt

min(t,r)

0 ds=

r 0 dt

t 0 ds+

a r dt

r 0 ds=

r 0 ds

a

s dt (126)

By solving the above Abel equation twice, we obtain the desired solution ϕ(t) = 2Q

π

a²−t², (127)

which is inserted into (121), yielding A(ξ) =Q

2 π

a^3/2J_3/2(ξa)

√ξ , (128)

where we have employed the result (Gradshteyn and Ryzhik, 1980)

π/2

0 sin²θJ1(aξsinθ)dθ= π

2aξJ_3/2(aξ). (129)

With the above obtained result (128), the entire electroelastic field of a cracked piezo- electric body can be determined. This can be achieved by substituting (128) into (105) for elastic displacements and potential. Making use of some known results involving infinite integrals of Bessel functions (Fabrikant, 2003; Li and Lee, 2004d)

∞

0 e^−cξJ1(ξr)J_3/2(ξa)

ξdξ = 2l²₁ a²−l₁²

√2πa^3/2r

l²₂−l₁², (130a)

∞

0 e^−cξJ₁(ξr)J_3/2(ξa)√dξ

ξ = √ r 2πa^3/2

−l₁ r²

r²−l²₁+ sin⁻¹ l₁

r

, (130b)

∞

0 e^−cξJ₀(ξr)J_3/2(ξa)

ξdξ = √ 2 2πa^3/2

sin⁻¹

l₁ r

− a l₂²−a² l²₂−l²₁

, (130c)

∞

0 e^−cξJ₀(ξr)J_3/2(ξa)√dξ

ξ = √ 2 2πa^3/2

a²−l₁²−csin⁻¹ l₁

r

, (130d)

explicit expressions for the elastic displacements and potential are obtained as follows u_r(r, z) = Q

π 3 j=1

a_j

rsin⁻¹ l1j

r

−l1j

r

r²−l_1j²

+B₁r, (131a)

u_z(r, z) = 2Q π

3 j=1

η_3jγ_ja_j

a²−l²_1j −γ_jzsin⁻¹ l_1j

r

+B₃z, (131b)

φ(r, z) = 2Q π

3 j=1

η_4jγ_ja_j

a²−l²_1j −γ_jzsin⁻¹ l_1j

r

−E^∞z, (131c)

whereB₁ andB₃are given by (111) and (112), respectively, and l_1j = 1

2

(r+a)²+ (γ_jz)²−

(r−a)²+ (γ_jz)²

(132a) l_2j = 1

2

(r+a)²+ (γ_jz)²+

(r−a)²+ (γ_jz)²

. (132b)

In a similar fashion, from (106) and (107), we can further give a complete solution of elastic stresses, electric displacements, and electric fields in the entire space. Or rather, the

distribution of electroelastic field in the entire half-space is

σ_rr(r, z) = 2Q π

3 j=1

β_0ja_jh_2j+c₁₁−c₁₂ 2 a_jh_0j

, (133a)

σ_θθ(r, z) = 2Q π

3 j=1

β_5ja_jh_2j−c₁₁−c₁₂ 2 a_jh_0j

, (133b)

σzz(r, z) = 2Q π

3 j=1

β1jajh2j +σ0, (133c)

σ_rz(r, z) =−2Q π

3 j=1

β_2ja_jh_1j, (133d)

srr(r, z) = Q π

3 j=1

(−2h2j+h0j)aj+B1, (134a)

s_θθ(r, z) = Q π

3 j=1

a_jh_0j+B₁, (134b)

s_zz(r, z) = 2Q π

3 j=1

η_3jγ²_ja_jh_2j+B₃, (134c)

srz(r, z) =−Q π

3 j=1

(η3j + 1)γjajh1j, (134d)

D_r(r, z) =−2Q π

3 j=1

β_3ja_jh_1j, (135a)

D_z(r, z) = 2Q π

3 j=1

β_4ja_jh_2j+D₀, (135b)

Er(r, z) = 2Q π

3 j=1

η4jγjajh1j, (135c)

E_z(r, z) =−2Q π

3 j=1

η_4jγ_j²a_jh_2j+E^∞, (135d)

whereβij(j= 1,2,3, i= 0, ...,4)are defined as before, and

β_5j = (c₁₃η_3j+e₃₁η_4j)γ_j²−c₁₂, (136)

and

h_0j = l_1j r²

r²−l²_1j−sin⁻¹ l_1j

r

, (137a)

h_1j = l_1j²

a²−l_1j² r

l²_2j−l_1j²

, (137b)

h_2j = a

l_2j² −a²

l²_2j−l²_1j −sin⁻¹ l_1j

r

. (137c)

Therefore, explicit analytic expressions for the complete electroelastic field are given in terms of elementary functions. From the above results, explicit expressions for electroelas- tic field for an impermeable crack and for a conducting crack can be directly written out only if settingε¯= 0 andε¯= ∞,respectively. In addition, if imposing the piezoelectric coefficients vanish, the electric and elastic behaviors are uncoupled, and the corresponding elastic field reduces to the results relating to a penny-shaped crack embedded in an infinite transversely isotropic medium subjected to uniform tension at infinity.

From (77) and (78), we get immediately elastic displacements and potential in the crack plane, from the results (131)

u_r(r,0) = QH(r−a) π

3 j=1

a_j

rsin⁻¹ a

r − a

r

r²−a²

+rQH(a−r) 2

3 j=1

a_j+(c₃₃e₃₁−c₁₃e₃₃)E^∞−c₁₃σ₀

(c11+c12)c33−2c²₁₃ r (138a) u_z(r,0) = 2QH(a−r)

π

3 j=1

η_3jγ_ja_j

a²−r², (138b)

φ(r,0) = 2QH(a−r) πκ

3 j=1

η_4jγ_ja_j

a²−r². (138c)

Furthermore, because of h0j(r,0) =

−^π₂, r < a,

ra²

√r²−a²−sin⁻¹_a

r

, r > a, (139a)

h_1j =

√ r

a²−r², r < a,

0, r > a, (139b)

h_2j =

−^π₂, r < a,

√ a

r²−a² −sin⁻¹_a

r

, r > a, (139c)

from the results (133) in conjunction with (135), it follows that aside from the apparent conclusionσ_rz(r,0) = 0,other non-vanishing elastic stresses, electric displacement, and

electric field in the crack plane are

σ_zz(r,0) = 2σ0

π

√ a

r²−a² −sin⁻¹_a

r

+σ₀, asr > a,

0, asr < a, (140a)

s_zz(r,0) = 2Q

π

√ a

r²−a² −sin⁻¹_a

r 3

j=1η_3jγ_j²a_j+B₃, asr > a,

B₃−Q, asr < a, (140b)

D_z(r,0) = ₂

(^D0−D¯)

π

√ a

r²−a² −sin⁻¹_a

r

+D₀, asr > a,

D,¯ asr < a, (140c)

E_z(x,0) =

−^2Q_π

√ a

r²−a² −sin⁻¹_a

r 3

j=1η_4jγ_j²a_j+E^∞, asr > a,

1¯

εD,¯ asr < a. (140d)

Similar to the two-dimensional case, for a penny-shaped crack in the three-dimensional piezoelectric space, it is seen from (140a) thatσ_zz(r, z)forz= 0is independent of applied electric loading and material properties. Nevertheless, the distribution ofσ_zz(r, z)forz= 0 around the crack front is reliant on the material properties, which is apparently seen from (133c) sincea_j together withκare determined by combined loading including mechanical loading as well as electric loading. In contrast,D_z(r, z)around the crack front is dependent on applied mechanical loading no matter whetherz= 0or not, which is due to the fact that D¯ is controlled by applied mechanical loading, apart from electric loading. Furthermore, it is not difficult to find that the behavior ofE_z(r, z)is the same asD_z(r, z).In addition, it is also observed that all the electroelastic quantities exhibit a usual square-root singularity near the crack frontr=a,in accordance with the counterpart for a two-dimensional case.