ScienceDirect. Analogy of stress singularities analysis between piezoelectric materials and anisotropic materials

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1 Available online at Scienceirect Procedia Materials Science ( 4 ) th European Conference on Fracture (ECF) Analog of stress singularities analsis between piezoelectric materials and anisotropic materials Toru SASAKI a *, Toshimi KONO a, Taeshi TANE b a epartmeno of Mechanical Engineering, Nagaoa National College of Technolog, Nagaoa, Niigata, JAPAN b epartmeno of Mechanical Engineering, Kitausu National College of Technolog, Kitaushu, Fuuoa, JAPAN Abstract In recent ears, intelligent or smart structures and sstems have become an emerging new research area. The joint structures of piezoelectric and piezoelectric materials in intelligent structures are often used. Piezoelectric materials have been etensivel used as transducers and sensors due to their intrinsic direct and converse piezoelectric effects that tae place between electric fields and mechanical deformation. The are plaing a e role as active components in man branches of engineering and technolog. Then it is nown that stress singularit frequentl occurs at interface due to a discontinuit of materials. The stress singularit fields are one of the main factors responsible for debonding under mechanical or thermal loading. So man investigations of stress singularities of piezoelectric materials have been conducted until now, but its eperimental studies are not so much. In this paper, with view to establish eperimental evaluation method of piezoelectric stress singularities, analog of basic formulation between piezoelectric materials and anisotropic materials are shown. And the analsis of stress singularities in piezoelectric materials containing crac or wedge is performed. Net the analsis of stress singularities in anisotropic materials is performed. Then the analog of their analsis theor is shown. 4 Elsevier The Authors. Ltd. Open Published access under b Elsevier CC BY-NC-N Ltd. license. Selection and and peer-review under under responsibilit responsibilit of the of Norwegian the Norwegian Universit Universit of Science of Science and Technolog and Technolog (NTNU), (NTNU), epartment epartment of Structural of Structural Engineering. Kewords: Stress singularit, Piezoelectric materials, Anisotropic materials, Analog; * Corresponding author. Tel: ; Fa: address: trsasai@nagaoa-ct.ac.jp Elsevier Ltd. Open access under CC BY-NC-N license. Selection and peer-review under responsibilit of the Norwegian Universit of Science and Technolog (NTNU), epartment of Structural Engineering doi:.6/j.mspro

2 768 Toru Sasai et al. / Procedia Materials Science ( 4 ) Introduction Piezoelectric materials have become an important branch of modern engineering materials with the recent development of the intelligent materials and structures. There has been considerable research on the stress singularities of piezoelectric materials b using theoretical and numericall analsis method (Sosa,99; Chue, C.H., Chen, C.., ; Xu, X.L., Rajapase, R.K.N., ;etc). However its eperimental studies are not so much. For validation of their analsis model and inverse analsis, the establishment of eperimental evaluation method of stress singularities is required. In this paper, basic formulations and analsis method of singularities problem for piezoelectric materials are shown. And basic formulations and analsis method of singularities problem for anisotropic materials are shown. Then the analog of their analsis theor is shown. And the possibilities of application to evaluation method of stress singularities are discussed.. Basic formulation.. Basic equation for piezoelectric materials The constitutive equation of piezoelectric materials for plane problems is given as follows(sosa,99): a a b a a b. a b 66 E b c. E b b c where, are normal and shear strains, E are electric fields,, are normal and shear stresses, are electric i ij i i ij i displacements, and a, b, c are reduced material constants for piezoelectric material. Elastic equilibrium and ij ij ij Gauss s law are given b,,. Furthermore the strains and electric field components satisf the compatibilit relations E E,. The equilibrium equations ma be satisfied b introducing to the stress functions U (, ) : U U U,,. In addition, we introduce an induction function (, ) such that,. which satisfies Gauss s law. Having to satisf the compatibilit relation, we obtain the following sstem the differential equations, U 4 4 ( ), ( ). (7) in which, and 4 are the differential operators of the second, third and fourth orders which have the form: () () () (4) (5) (6)

3 Toru Sasai et al. / Procedia Materials Science ( 4 ) c c, b b b, a a a a Eq.(7) is reduced to the single sith order differential equation as follows: U (9) 4 ( ) Eq. (9) can be solved b means of comple variables. Set U U ( z ) U ( ) and is a comple number, the characteristic equation of Eq.(9) is 6 4 a c a c a c a c b b b b a c a c a c () b b b b a c b. (8).. Basic equation for anisotropic materials The constitutive equation of anisotropic materials is given as follows(lehnitsii,98): where s ij given b z z z z s s i j s. () ij ij s are elastic compliance constants, ij, z, z. Furthermore the strains satisf the compatibilit relations. are reduced elastic compliance constants. Elastic equilibrium are () () (4) z z. The equilibrium equations ma be satisfied b introducing to the two stress functions F (, ), (, ) : F F F,,. (5) (6) z,. z Having to satisf the compatibilit relation, we obtain the following sstem the differential equations, L F L, L F L. (9) 4 in which L, L and L 4 are the differential operators of the second, third and fourth orders which have the form: (7)

4 77 Toru Sasai et al. / Procedia Materials Science ( 4 ) L s s s, L s ( s s ) ( s s ) s, L s s ( s s ) s s Eq.(9) is reduced to the single sith order differential equation as follows: ( L L L ) F. () 4 Eq. () can be solved b means of comple variables. Set the characteristic equation of Eq.() is ˆ ˆ ˆ 4 F F ( z ) F ( ˆ ) () and ˆ is a comple number, l ( ) l ( ) l ( ). () ˆ ˆ ˆ l ( ) s s s, l ( ˆ ) s ˆ ( s s ) ˆ ( s s ) ˆ s, l ( ˆ ) s ˆ s ˆ ( s s ) ˆ s ˆ s ().. Analg of basic formulation The solution of Eqs.(9) can be written b as (4) U Re U ( z ), Re U ( z ). z where z (,,) and comple constants are as follows; ( b b ) b c c, (,, ). (5) And the solution of Eqs.() can be written b as F (, ) Re F ( z ), where comple constants ˆ are as follows; 4 (, ) Re ˆ F ( z ). (6) ˆ l ( ˆ ) l ( ˆ ) l ( ˆ ) l ( ˆ ) (,,). (7) Then b introducing new function which are defined as ( z ) U ( z ) F ( z ), (,,). (8) z the components of stress, electric displacement, displacement, electric potential etc. are derived as Table.. Table. show that the components of in-plane stress etc. are identical formulation, components of electric displacement and out-of-plane stress are analogical formulation. These analogies indicate that eperiment of piezoelectric materials can be replaced b eperiment of anisotropic elastic materials. It is ver useful because eperiment of piezoelectric materials is ver sensitive for its environment and specimen.. Analog of stress singularities analsis.. Crac problem where In the crac problem,we see the epressions for the function in the following form(lehnitsii,98): ( z ) A a A b A c. m () m m m det A m A are cofactors of the following matri. ij

5 Toru Sasai et al. / Procedia Materials Science ( 4 ) A And the coefficient a, b, c are determined from the boundar condition. m m m () Table. Basic formulations of piezoelectric material and anisotropic elastic material. Piezoelectric Material Anisotropic Elastic Material <In-Plane Stress> <In-Plane Stress> Re ( z ), Re ˆ ( z ), Re ( z ), Re ( z ). <Electric isplacement> Re ( z ), Re ( z ). <Comple Constants> p a a b, q a a b, r b c, (,,). <In-Plane Resultant Stress> P Re ( z ), P Re ( z ). <In-Plane Resultant Electric isplacement> Re ( ). n z <In-Plane isplacement> u Re p ( z ) u, v Re q ( z ) v. <Electric Potential> Re r ( z ). Re ( z ), Re ˆ ( z ), <Out-of-Plane Stress> Re ˆ ˆ ( z ), z Re ˆ ( z ). z <Comple Constants> pˆ { s ˆ s s ˆ ( s ˆ s )}, ˆ qˆ { s ˆ s s ˆ ( s ˆ s )}, ˆ rˆ { s ˆ s s ˆ ( s ˆ s )}, ˆ (,,). <In-Plane Resultant Stress> P Re ˆ ( z ), P Re ( z ), <Out-of-Plane Resultant Stress> P Re ˆ ( ). z z <In-Plane isplacement> u Re pˆ ( z ) u, v Re rˆ ( z ) v, <Out-of-Plane isplacement> w Re rˆ ( z ).

6 77 Toru Sasai et al. / Procedia Materials Science ( 4 ) For piezoelectric materials, boundar condition are given as Traction free : P, P, Electricall open :. n on crac. And for anisotropic materials, boundar condition are given as Traction free : P, P, P, on crac... Wedge problem (5) z In the wedge problem, we see the epressions for the function in the following form(chue, C.H., Chen, C.., ): ( z ) C z z where is eigenvalue. The coefficient C,, are determined from the boundar condition. For piezoelectric materials, boundar condition are given as Traction free : P P, Electricall open :. n on edge Continuit of stress and electric displacement : P P, P P,, n n Continuit of displacement and electric potential : u u, v v,. And for anisotropic materials, boundar condition are given as Traction free : P P P, on edge. z on interface (8) (4) (6) (7) Continuit of stress: P P, P P,, n n Continuit of displacement : u u, v v, w w. on interface (9) 4. Conclusions Analogies of basic formulation and governing equation between piezoelectric materials and anisotropic materials were shown. The components of stress, electric displacement, displacement, electric potential etc. are derived b using these analogies. Analtical methods for crac and wedge problem were derived b unifing formulation. In the future wor, we will establish eperimental evaluation method of piezoelectric stress singularities b using these analogies. Acnowledgements This wor was supported b JSPS KAKENHI Grant Number 584. References Horacio Sosa, 99. On the fracture mechanics of piezoelectric solids. International Journal of Solids and Structures 9, 6 6. Chue, C.H., Chen, C..,. Antiplane stress singularities in a bonded biomaterial piezoelectric wedge. Archive of Applied Mechanics 7, Xu, X.L., Rajapase, R.K.N.,. On singularities in composite piezoelectric wedges and junctions. International Journal of Solids and Structures 7, Tong-Yi Zhang, Pin Tong, 996. Fracture mechanics for a mode III crac in a piezoelectric material. International Journal of Solids and Structures, Lehnitsii, S.G., 98. Theor of Elasticit of an Anisotropic Bod, Mir Publishers, Moscow.