Numerical Results and Discussion

The electroelastic singularity is governed by the real part of (λ -1), and the root of _m primary interest is the one with the smallest positive real part between zero and one. In this section, the values of minimum Re[λ ] are shown for single material and bi-material bodies _m of revolution. The piezoelectric materials, PZT-4 and PZT-5H, and an elastic material, Al (aluminium), are considered. The material properties of PZT-4 and PZT-5H are given in Table 1, while the elastic constants for Al are E (Young’s modulus) = 68.9 GPa and ν (Poisson’s ratio) = 0.25. The results were obtained using eight equal sub-domains for ϕ and 15-term series solutions for each sub-domain. The boundary conditions under consideration are specified by four letters. The first pair of letters refers to the boundary conditions at ϕ =ϕ₀, while the second pair specifies the boundary conditions at ϕ =ϕ_n. The first letter in each pair concerns the mechanical boundary conditions, with C and F’s denoting clamped and free boundary conditions, respectively, while the second letter concerns the electric boundary conditions with C and O’s representing electrically closed and open boundary conditions, respectively. Accordingly, in the following, COFO boundary conditions mean that the mechanical boundary conditions are clamped and free at ϕ =ϕ₀ and ϕ =ϕ_n, respectively, and the electric boundary conditions are open at ϕ =ϕ₀ and ϕ =ϕ_n.

3.4.1 Bodies of revolution made of a single piezoelectric material

Consider a PZT-4 or PZT-5H body of revolution with a direction of polarization that may not be along the Z-axis (axis of revolution). The geometry of the body considered in this section is similar to geometry II in Fig. 3.4. Figures 3.5 plots the variations of minimum Re[λ ] with _m θ for PZT-4 bodies with γ =0^, 45^ and 90^, while Fig. 3.6 plots corresponding curves for PZT-5H bodies. Notably, the results at θ =2π −θ₀ are identical to those at θ = in all the cases that are considered in this work. Consequently, the range of θ₀ θ considered is between 0^ and 180^. The λ that corresponds to minimum Re[_m λ ] are _m all real in the cases examined in Figs. 3.5 and 3.6.

As expected, minimum Re[λ ] does not change with _m θ when the direction of polarization is along the Z-axis (γ =0^). When the direction of polarization is not along the Z-axis, minimum Re[λ ] varies significantly with θ . For example, when _m γ =45^, the maximum relative difference may reach 7.8% for a PZT-4 body with COCO boundary conditions, while the maximum difference is about 5.2 % for a PZT-5H body. When γ changes from 0^ to 45^ or 90^, the minimum Re[λ ] may increase or decrease, _m

31

depending on the values of θ and the boundary conditions. PZT-4 bodies exhibit more severe electroelastic singularities than PZT-5H bodies under clamped-clamped mechanical boundary conditions; the opposite is true under free-free mechanical boundary conditions.

Figures 3.7 and 3.8 display the variations in minimum Re[λ ] at _m θ =60^with β for PZT-4 and PZT-5H bodies, respectively. Two values of γ , 0^and 45^, were considered.

Generally, minimum Re[λ ] declines as β increases, such that a larger β induces more _m severe electroelastic singularities at the sharp corner of a body of revolution. Electroelastic singularities under free-free boundary conditions are more severe than those obtained under clamped-clamped boundary conditions. When γ =0^, the electric boundary conditions do not significantly affect the singularities. However, when γ =45^ , open-open electric boundary conditions results in a smaller minimum Re[λ ] than closed-closed electric _m boundary conditions for clamped-clamped bodies of revolution, while the opposite trend is true for bodies of revolution with free-free mechanical boundary conditions. As γ changes from 0^ to 45^, the λ , which corresponds to minimum Re[_m λ ], may change _m from real to complex or from complex to real. For instance, under CCCC boundary conditions, λ_m are complex for γ =45^ when β is between 48^ and 73^ for PZT-4 bodies and between 52^ and 70^ for PZT-5H bodies, while they are all real for γ =0^. A comparison of Figs. 3.7 and 3.8 reveals that PZT-4 bodies have stronger singularities than PZT-5H bodies under clamped-clamped boundary conditions, but not at all values of β under free-free boundary conditions

3.4.2 Bi-material bodies of revolution made of piezoelectric and elastic materials

This section investigates bi-material bodies of revolution with a geometry that is similar to geometry II in Fig. 3.4, in which material 1 is an isotropic elastic material, Al, and material 2 is PZT-4 or PZT-5H. The arrangements considered in Figs. 3.9 to 3.12 are the same as those in Figs. 3.5 to 3.7, respectively, except that bi-material bodies of revolution are considered in Figs. 3.9 to 3.12. Notably, the continuity conditions on the interface between the piezoelectric material and the elastic material are given by Eqs. (3.9a) to (3.9c), (3.9e) to (3.9g) and

( )ⁱ⁺¹ (r,θ,ϕ_i)=0

φ . No electric boundary condition applies at ϕ =ϕ₀, and the second letter of the four letters that denote the boundary conditions is replaced by “-“.

Figures 3.9 and 3.10 discover that when γ ≠0^, the minimum Re[λ ] does significantly _m vary with θ . When γ =45^, the maximum relative difference may reach 25% for a PZT-4/Al body under C-CC boundary conditions, while the maximum difference is approximately 16 % for a PZT-5H/Al body. Unlike in Figs. 3.5 and 3.6, the minimum Re[λ ] _m for the C-CC boundary conditions can be smaller than those for C-CO boundary conditions, depending on γ and θ . When γ =45^, the λ , which correspond to minimum Re[_m λ ] _m under C-CC boundary conditions, are no longer all real; they are complex for 63^ ≤θ ≤110^ in Fig. 3.9(b) and 68^ ≤θ ≤101^ in Fig. 3.10(b). Figure 3.9(b) demonstrates that the minimum Re[λ ] under C-CC boundary conditions are lower than those under free-free _m boundary conditions when θ ≤14^.

Figures 3.11 and 3.12 plot the variations of minimum Re[λ ] at _m θ =60^with β for PZT-4/Al and PZT-5H/Al bodies, respectively. The relatively abrupt changes in the curves (i.e., atβ ≈159^ under F-FC boundary conditions and β ≈99^ under C-CO boundary

conditions in Fig. 3.11(a)) are caused by the roots’s changing from real to complex numbers or from complex to real numbers. Generally, the strength of the electroelastic singularity increases with β . Free-free boundary conditions produce singularities that are more severe than clamped-clamped boundary conditions do, except for β ≥160^ . Interestingly, the minimum Re[λ ] for the bodies with _m γ =0^are more considerably affected by electric boundary conditions than those for the bodies with γ =45^. Changing γ from 0^ to 45^ can alter the minimum Re[λ ] with the maximum relative difference of 9.6% occurring at _m

99

β = under C-CO boundary conditions in Fig. 3.11. Unlike the minimum Re[λ ] ≥ 0.5 _m for bodies of revolution made of two isotropic elastic materials under free-free boundary conditions given in Huang and Leissa [26], the minimum Re[λ ] can be smaller than 0.5 for _m

β larger than around 150^under F-FO boundary conditions.

3.4.3 Bi-material bodies of revolution made of piezoelectric materials

The results for bi-material bodies of revolution consisting of PZT-4 and PZT-5H with a horizontal interface are given in Figs. 3.13 and 3.14. Figure 3.13 concerns bodies of revolution with geometry I and geometry II displayed in Fig. 3.4, where materials 1 and 2 are PZT-5H and PZT-4, respectively. Figure 3.14 considers bodies of revolution with geometry II and having various β .

As expected, Fig. 3.13 demonstrates that bodies of revolution with geometry II (α =270^) have more severe singularities at the interface corner than do bodies of revolution with geometry I (α =180^). When γ =0^, the roots corresponding to minimum Re[λ ] are all _m real. As γ changes from 0^ to 45^ or 90^, the roots may change from real to complex, depending on θ and the boundary conditions. For instance, for γ =45^and under COCO boundary conditions, when θ <51^ and θ <36^ for geometries I and II, respectively, the roots corresponding to minimum Re[λ ] are complex. The variations of minimum Re[_m λ ] _m with θ in Fig. 3.13(b) indicate that the maximum difference can reach 11% for geometry I under FOFO boundary conditions, and 7.2% for geometry II under FOFO boundary conditions. When γ =90^, the maximum difference between values of minimum Re[λ ] for _m various θ reaches 4.5% for geometry I under COCO boundary conditions, and 4.3% for geometry II under FOFO boundary conditions.

Figure 3.14 plots the variations of minimum Re[λ ] at _m θ =60^ with β for bodies of revolution with geometry II. Two values of γ , 0^and 45^, were considered. Again, the relatively abrupt changes in the curves are caused by a change in the roots from real to complex or from complex to real. Generally, free-free boundary conditions give more severe singularities at the interface corner than do clamped-clamped boundary conditions. Changing γ from 0^ to 45^ changes the minimum Re[λ ] by up to 5.0%, as for the body of _m revolution with β =105^ under COCO boundary conditions.

33

IV Concluding Remarks

This study found asymptotic solutions to piezoelectric wedges and bodies of revolution to investigate geometrically-induced electroelastic singularities in these bodies based on three-dimensional piezoelaticity theory in a cylindrical coordinate system. The piezoelectric material is first assumed to be anisotropic and its direction of polarization to be arbitrary.

The solutions were obtained using an eigenfunction expansion approach in conjunction with a power series technique to solve the equilibrium and Maxwell’s equations, which are four coupled partial differential equations in terms of the displacement components and electric potential. The present solutions are easily reduced to the solution for anisotropic elastic wedges by eliminating the piezoelastic and dielectric constants. The proposed solutions are verified by performing convergence studies and comparing the results with the published results.

The proposed solution were employed to examine electroelastic singularities in wedges and bodies of revolution that comprise a single piezoelectric material, bounded piezo/isotropic elastic materials, or piezo/piezo materials. The minimum Re[λ ], which is _m directly related to the order of the singularity, is displayed for different corner angles, combinations of boundary conditions, and directions of polarization. As expected, the strength of the singularity generally increases with the increase of corner angle. The geometrically induced electroelastic singularity order can depend significantly on the polarized direction. Interestingly, the direction of polarization can be set to eliminate the singularities at the interface of 180^owedges made of PZT-5H/Si or PZT-5H/PZT-4 with free-free mechanical boundary conditions. This phenomenon is particularly important because such wedges are frequently encountered in many smart structures.

Appendix I

35

Appendix II

e z

where the electric potential, φ , is related to the electric field by,

E_r r

∂

= ∂φ

, θ

φ

θ ∂

= ∂ E 1r

and E_z z

∂

=∂φ .

37

( )

22 24

9 θ η^e

s =− ,

( )

_^

( )

_^

∂ +∂ +

−

= λ θ

θ η _{14 25} ²⁴

22 10

1 e

e e

s _m ,

( )





 





∂ +∂

−

= λ λ θ

θ η ₁₅ ²⁵

22 11

1 e

e

s _{m m}

39

V. References

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Burton, W. S., Sinclair, G. B., 1986. On the singularities in Reissner’s theory for the bending of elastic plates. Journal of Applied Mechanics, ASME 53, 220-222.

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Chen, T. H., Chue, C. H., Lee, H.T., 2004. Stress singularities near the apex of a cylindrically polarized piezoelectric wedge. Archieve of Applied Mechanics 74, 248-261.

Chue, C. H., Chen, C. D., 2002. Decoupled formulation of piezoelectric elasticity under generalized plane deformation and its application to wedge problems. International Journal of Solids and Structures 39, 3131-3158.

Chue, C. H., Chen, C. D., 2003. Antiplane stress singularities in a bonded bimaterial piezoelectric wedge. Archieve of Applied Mechanics 72, 673-685.

Dempsey, J. P., Sinclair, G. B., 1981. On the stress singular behavior at the vertex of a bi-material wedge. Journal of Elasticity 11, 317-327.

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Hein, V. L., Erdogan, F., 1971. Stress singularities in a two-material wedge. International Journal for Fracture Mechanics 7, 317-330.

Huang, C. S., 2002a. Corner singularities in bi-material Mindlin plates. Composite Structures 56, 315-327.

Huang, C. S., 2002b. On the singularity induced by boundary conditions in a third-order thick plate theory. Journal of Applied Mechanics, ASME 69, 800-810.

Huang, C. S., 2003. Stress singularities in angular corners in first-order shear deformation plate theory.International Journal of Mechanical Science 45, 1-20.

Huang, C. S., Leissa, A.W., 2007. Three-dimensional sharp corner displacement functions for bodies of revolution. Journal of Applied Mechanics, ASME 74, 41-46.

Huang, C. S., Leissa, A. W., 2008. Stress singularities in bimaterial bodies of revolution.

Composite Structures 82(4), 488-498.

Hwu, C., Ikeda, T., 2008. Eletromechanical fracture analysis for corners and cracks in piezoelectric materials, International Journal of Solids and Structures 45, 5744-5764.

Li, Y. L., Hu, S. Y., Munz, D., Yang, Y. Y., 1998. Asymptotic description of the stress field around the bond edge of a cylindrical joint. Archieve of Applied Mechanics 68(7), 552-565.

Li, Y. L., Hu, S. Y., Yang, Y. Y., 2000. Stresses around the bond edges of axisymmetric deformation joints. Fracture Mechanics 66(2), 153-170.

Li, Y., Sato, Y., Watanabe, K., 2002. Stress singularity analysis of axisymmetric piezoelectric bonded structure. JSME International Journal Series A 45(3), 363-370.

Love A. E. H. A, 1927. Treatise on the Mathematical Theory of Elasticity, 4^th ed., The Macmillan Co, New York.

Müller, D.E., 1956. A method for solving algebraic equations using an automatic computer.

Mathematical Tables and Aids to Computation 10, 208-215.

McGee, O.G., Kim, J.W., 2005. Sharp corner functions for Mindlin plates. Journal of Applied Mechanics, ASME 72(1), 1-9.

Saidi, A.R., Hejripour, F., Jomehzadeh, E., 2010. On the stress singularities and boundary layer in moderately thick functionally graded sectorial plates. Applied Mathematical Modeling 34(11), 3478-3492.

Shang, F., Kitamura, T., 2005. On stress singularity at the interface edge between piezoelectric thin film and elastic substrate, Microsystem Technology 11, 1115-1120.

Shang, F., Kitamura, T., Hirakata, H., Kanno, I., Kotera, H. M., Terada, K., 2005.

Experimental and theoretical investigations of delamination at free edge of interface between piezoelectric thin films on a substrate, International Journal of Solids and Structures 42, 1729-1741.

Sosa, H. A., Pak, Y. E., 1990. Three-dimensional eigenfunction analysis of a crack in a piezoelectric material. International Journal of Solids and Structures 26, 1-15.

Sze, K. Y., Wang, H. T., Fan, H., 2001. A finite element approach for computing edge singularities in piezoelectric materials, International Journal of Solids and Structures 38, 9233-9252.

Timoshenko, S. P., Goodier, J. N., 1970. Axisymmetric stress and deformation in a solid of revolution. In: Theory of elasticity, 3rd ed. Kogakusha: McGraw-Hill, 428-429.

Ting. T. C. T., Jin, Y., Chou, S. C., 1985. Eigenfunctions at a singular point in transversely isotropic materials under axisymmetric deformations. ASME Journal of Applied Mechanics 52(3), 565-569.

Wang, Z., Zheng, B., 1995. The general solution of three dimensional problems in piezoelectric media, International Journal of Solids and Structures 32, 105-115.

Williams, W. L., 1952a. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. Journal of Applied Mechanics, ASME 19, 526-528.

Williams, W. L., 1952b. Stress singularities resulting from various boundary conditions in

41

angular corners of plates under bending. Proceedings of 1^st U. S. National Congress of Applied Mechanics, ASME, New York, 325-329.

Williams, M.L., Owens, R.H., 1954. Stress singularities in angular corners of plates having linear flexural rigidities for various boundary conditions. Proceeding of 2^nd U.S. National Congress of Applied Mechanics, 407-411.

Xu, J. Q., Mutoh, Y., 2001. Singularity at the interface edge of bonded transversely isotropic piezoelectric dissimilar materials. JSME International Journal Series A 44(4), 556-566.

Xu, X. L., Rajapakse, R. K. N. D., 2000. On singularities in composite piezoelectric wedges and junctions. International Journal of Solids and Structures 37, 3235-3275.

Ying, X., Katz, I. N., 1987. A uniform formulation for the calculation of stress singularities in the plane elasticity of a wedge composed of multiple isotropic materials. Computers and Mathematics with Applications 14, 437-458.

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Journal of Applied Mechanics, ASME 31, 150-152.

Table 2.1 Material properties

Material

Stiffness [GPa]

Piezoelectric const.

[C/m²]

Dielectric const.

×10^-10[F/m]

ˆ11

c cˆ₁₂ cˆ₁₃ cˆ₃₃ cˆ₄₄ ˆe₁₅ ˆe₃₁ ˆe₃₃ ηˆ₁₁ ηˆ₃₃

CdSe 74.1 45.2 39.3 83.6 13.2 -0.138 -0.159 0.347 0.844 0.903

PZT-4 139.0 77.8 74.3 115.0 25.6 12.7 -5.2 15.1 64.6 56.2

PZT-5H 126.0 55.0 53.0 117.0 35.3 17.0 -6.5 23.3 151.0 130.0

BaTiO₃ 275.0 179.0 152.0 165.0 54.3 21.3 -2.69 3.65 175.0 9.88

PZT-6B(Im.) 168.0 60.0 60.0 163.0 27.1 43.0 -14.0 36.0 200.0 247.0

PZT-6B 168.0 60.0 60.0 163.0 27.1 4.6 -0.9 7.1 36.0 34.0

Si 166.2 64.6 64.6 166.2 50.8 - - - - -

43

Table 2.2 Convergence of minimum Re[λ ] for PZT-4 wedges _m

γ Boundary conditions

Number of Sub-domains

Terms Published

results

5 6 7 8 9 10 12 14 15

360^ FOFO

3 0.4985 0.4978 0.4916 0.4417 0.4980 0.4998 0.4750 0.5000 0.4999

0.5000^# 4 0.4996 0.4963 0.4993 0.4999 0.4999 0.4984 0.4999 0.4999 0.4999

6 0.5000 0.4993 0.4999 0.4999 0.5000 0.5000 0.4999 0.5000 0.5000 8 0.4999 0.4999 0.4999 0.4999 0.4999 0.5000 0.5000 0.4999 0.5000

360^

FOCC

3 0.1769 0.1969 0.2052 0.1602 0.1718 0.1965 0.1724 0.1954 0.1895

0.1869*

4 0.1855 0.1895 0.1847 0.1877 0.1879 0.1857 0.1877 0.1865 0.1869 6 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869 8 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869

180^

2 0.3710 0.3751 0.3749 0.3740 0.3736 0.3735 0.3741 0.3737 0.3738

0.3739*

3 0.3741 0.3737 0.3738 0.3739 0.3739 0.3739 0.3739 0.3739 0.3737 4 0.3738 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739 6 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739 Note: * denotes results from Hwu and Ikeda (2008)

# denotes results from Sosa and Pak (1990)

Table 2.3 Comparisons between the present and the published λ for PZT-4 wedges _m

γ ^Boundary

conditions

Direction of polarization

Roots of

λm Published results Present results

360^ FOCC Y

λ0

λ1

λ2

0.1869^* 0.3131^* 0.6869^*

0.1869 0.3131 0.6869

357^ FOFO

Y

λ0

λ1

λ2

0.5000^# 0.5094^# 0.5046^#

0.5000 0.5094 0.5046

Z

λ0

λ1

λ2

0.5000^# 0.5085^# 0.5042^#

0.5000 0.5085 0.5042

330^ FOFO

Y

λ0

λ1

λ2

0.5021^# 0.5499^# 0.6109^#

0.5021 0.5498 0.6109

Z

λ0

λ1

λ2

0.5015^# 0.5455^# 0.5982^#

0.5014 0.5455 0.5981

180^ FOCC Y

λ0

λ1

λ2

0.3739^* 0.5000^* 0.6261^*

0.3739 0.5000 0.6261 Note: * denotes the results of Hwu and Ikeda (2008)

#: denotes the results of Sze et al. (2001)

45

Table 3.1: Convergence of minimum Re[λm] for bodies of revolution

Geometry

Material 1/

Material 2

Number of Sub-domains

Number of Polynomial terms Published

results

5 6 7 9 11 13 15

I

CdSe/

PZT-5H

2 0.9363 0.9348 0.9357 0.9377 0.9387 0.9383 0.9380

0.9381*

4 0.9379 0.9381 0.9382 0.9381 0.9381 0.9381 0.9381 6 0.9381 0.9381 0.9381 0.9381 0.9381 0.9381 0.9381 8 0.9381 0.9381 0.9381 0.9381 0.9381 0.9381 0.9381

CdSe/

PZT-6B

2 0.9268 0.9242 0.9308 0.9302 0.9280 0.9272 0.9278

0.9281*

4 0.9286 0.9289 0.9279 0.9281 0.9281 0.9281 0.9281 6 0.9281 0.9281 0.9281 0.9281 0.9281 0.9281 0.9281 8 0.9281 0.9281 0.9281 0.9281 0.9281 0.9281 0.9281

CdSe/ BaTiO3

2 0.8949 0.9588 0.9429 0.9172 0.9394 0.9256 0.9284

0.9429*

4 0.9436 0.9430 0.9430 0.9430 0.9429 0.9429 0.9429 6 0.9429 0.9428 0.9428 0.9429 0.9429 0.9429 0.9429 8 0.9429 0.9428 0.9429 0.9429 0.9429 0.9429 0.9429

PZT-6B/

PZT-6B(Im.)

2 0.98792 0.98475 0.98641 0.98793 0.98713 0.98828 0.98792

0.98724⁺ 4 0.98742 0.98732 0.98731 0.98720 0.98613 0.98725 0.98724

6 0.98802 0.98764 0.98764 0.98733 0.98724 0.98724 0.98724 8 0.98730 0.98724 0.98723 0.98724 0.98724 0.98724 0.98724

II

PZT-6B/

PZT-6B(Im.)

3 0.54766 0.53669 0.52792 0.52053 0.52716 0.52670 0.53197

0.52819⁺ 6 0.52694 0.52758 0.52801 0.52836 0.52819 0.52818 0.52820

9 0.52803 0.52809 0.52823 0.52820 0.52820 0.52820 0.52820

Note: * denotes the results of Sato and Watanabe (2002) +: denotes the results of Xu and Mutoh (2001)

Fig. 2.1 Coordinate systems for a wedge

47

Fig. 2.2 Sub-domains for θ∈[0, ]γ

(a) (b)

(c)

Fig. 2.3 Variation of minimum Re[λ ] with direction of polarization for a _m 270^o PZT-5H wedge (a) α =0^o (on x-z plane), (b) β =90^o (on x-y plane), (c) β =30 , 60 and 90^{o o o}

PZT-5H

PZT-5H

PZT-5H

49

(a)

(b)

Fig. 2.4 Variation of minimum Re[λ ] with wedge angle for PZT-5H wedges _m (a) FOFO boundary conditions,

(b) COCO boundary conditions

PZT-5H

(a) (b)

(c)

Fig. 2.5 Variation of minimum Re[λ ] with direction of polarization for a _m 180^oPZT-5H/ Si bi-material wedge (a) α =0^o (on x-z plane), (b) β =90^o (on x-y plane), (c) β =30 , 60 and 90^{o o o}

Si PZT-5H

Si PZT-5H

Si PZT-5H

51

(a) (b)

(c)

Fig. 2.6 Variation of minimum Re[λ ] with direction of polarization for a _m 270^oPZT-5H/ Si bi-material wedge (a) α =0^o (on x-z plane), (b) β =90^o (on x-y plane), (c) β =30 , 60 and 90^{o o o}

Si PZT-5H

Si PZT-5H

PZT-5H Si

(a)

(b)

Fig. 2.7 Variation of minimum Re[λ ] with wedge angle for PZT-5H/Si wedges _m (a) F-FO boundary conditions,

(b) C-CO boundary conditions

Si PZT-5H

53

(a) (b)

(c)

Fig. 2.8 Variation of minimum Re[λ ] with direction of polarization for a _m 180^oPZT-5H/ PZT-4 bi-material wedge

(a) α =0^o (on x-z plane), (b) β =90^o (on x-y plane), (c) β =30 , 60 and 90^{o o o}

PZT-4 PZT-5H

PZT-4 PZT-5H

PZT-4 PZT-5H

(a) (b)

(c)

Fig. 2.9 Variation of minimum Re[λ ] with direction of polarization for a _m 270^oPZT-5H/ PZT-4 bi-material wedge

(a) α =0^o (on x-z plane), (b) β =90^o (on x-y plane), (c) β =30 , 60 and 90^{o o o}

PZT-4 PZT-5H

PZT-4 PZT-5H

PZT-5H PZT-4

55

(a)

(b)

Fig. 2.10 Variation of minimum Re[λ ] with wedge angle for PZT-5H/PZT-4 wedges _m (a) FOFO boundary conditions,

(b) COCO boundary conditions

PZT-4 PZT-5H

Fig. 3.1 Bi-material body of revolution with a sharp corner

57

Fig. 3.2 Cylindrical (r, Z) and sharp corner (ρ,φ) coordinates

Fig. 3.3 Sub-domains forϕ∈

[

ϕ₀,ϕn

]

59

Fig.3.4 Geometry and boundary conditions for bodies of revolution considered in convergence studies

Material 2

Material 1

Material 2

Material 1

α =180° α =270°

Geometry I Geometry II

Free + Opened

Free + Opened

Free + Opened

Z Z

β= 90°

(a)

(b)

(c)

Fig. 3.5 Variation of minimun Re[λm] with θ for PZT-4 of bodis of revolution with β = 90°: (a) γ = 0°, (b) γ = 45°, (c) γ = 90°

PZT-4 β=90∘

61

(a)

(b)

(c)

Fig. 3.6 Variation of minimun Re[λm] with θ for PZT-5H of bodis of revolution with β = 90°: (a) γ = 0°, (b) γ = 45°, (c) γ = 90°

PZT-5H β=90∘

(a)

(b)

Fig. 3.7 Variation of minimun Re[λm] at θ = 60° with β for PZT-4 bodies of revolution:

(a) γ = 0°, (b) γ = 45°

PZT-4 β

63

(a)

(b)

Fig. 3.8 Variation of minimun Re[λm] at θ = 60° with β for PZT-5H bodies of revolution:

(b) γ = 0°, (b) γ = 45°

PZT-5H β

(a)

(b)

(c)

Fig. 3.9 Variation of minimun Re[λm] with θ for PZT-4/Al of bodis of revolution with β = 90°: (a) γ = 0°, (b) γ = 45°, (c) γ = 90°

Al β=90∘

PZT-4

65

(a)

(b)

(c)

Fig. 3.10 Variation of minimun Re[λm] with θ for PZT-5H/Al of bodis of revolution with β= 90°: (a) γ = 0°, (b) γ = 45°, (c) γ = 90°

Al β=90∘

PZT-5H

(a)

(b)

Fig. 3.11 Variation of minimun Re[λm] at θ = 60° with β for PZT-4/Al bodies of revolution:

(a) γ = 0°, (b) γ = 45°

Al PZT-4

β

67

(a)

(b)

Fig. 3.12 Variation of minimun Re[λm] at θ=60° with β for PZT-5H/Al bodies of revolution: (a) γ

= 0°, (b) γ = 45°

Al PZT-5H

β

(a)

(b)

(c)

Fig. 3.13 Variation of minimun Re[λm] with θ for PZT-4/ PZT-5H of bodis of revolution with β = 90°: (a) γ = 0°, (b) γ = 45°, (c) γ = 90°

α=180∘

PZT-5H β=90∘

α=270∘

PZT-5H β=90∘

PZT-4 PZT-4

69

(a)

(b)

Fig. 3.14 Variation of minimun Re[λm] at θ=60° with β for PZT-4/ PZT-5H bodies of revolution:

(a) γ = 0°, (b) γ = 45°.

PZT-5H PZT-4

β

在文檔中 壓電材料迴轉體與楔形板幾何引致電彈奇異性之探討 (頁 34-73)