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Electromechanical Linear Contribution at Fracture of Crack on Pre-notched Piezoelectric Ceramics Under Double Torsion

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Abstract

The linear contribution of total energy release rate for fracture of ferroelectric materials was studied based on the linear theoretic framework of piezoelectricity using the double-torsion technique. It was found that three linear contributions of the total energy release rate for piezoelectric material are definitely associated with pure mechanical compliance, electromechanical compliance, and pure electrical capacitance. To confirm the entity of contributions, mechanical and electromechanical compliances were analyzed obtained from result of load vs. displacement measured on each electric field, and electrical capacitance was obtained from property and geometry of samples.

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Abbreviations

a :

Crack length

a 0 :

Smaller crack length than a

A :

Crack surface area

b :

Notch length from edge of specimen to the loading and supporting line

c L , c T , c A :

Constants of Irwin matrix

c E ij or \( {\overline{c}}_{ij}^E \) :

Elastic stiffness constants measured at a constant electric field

C F e :

Pure electric capacitance

C V m :

Pure mechanical compliance

C E :

Conventional electric capacitance

C M :

Conventional mechanical compliance

C p :

Piezoelectric compliance

\( {\overline{C}}_{xx}^{*} \) :

Shear modulus of transversely isotropic piezoelectric material

\( {\overline{d}}_{ij} \) :

Piezoelectric strain constants

D ii :

Electric displacement components

e :

Constant of Irwin matrix

e ij or ē ij :

Piezoelectric stress constants

E 0 or E :

Applied electric field

E ii :

Electric field components

F :

Applied mechanical force

F int :

Fracture initiation load

G :

Crack driving force or energy release rate

G m :

Pure mechanical contribution of total energy release rate

G e :

Pure electric contribution of total energy release rate

G M :

Conventional mechanical contribution of total energy release rate

G E :

Conventional electric contribution of total energy release rate

G p :

Piezoelectric contribution of total energy release rate

G intr :

Intrinsic toughness

G tip l :

Crack-tip energy release rate

G tot l or G tot :

Total energy release rate in linearly piezoelectric assumption

[H]:

Irwin matrix

K I , K II  , K III :

Mechanical crack tip stress intensity factors

K IV (or K D ):

Crack tip electric displacement intensity factor

L :

Length of specimen

p i :

Eigenvalues

Q :

Electric charge

Q l :

Linear component of electric charge

Q E l :

Pure electrical contribution of linear component of electric charge

Q P l :

Piezoelectric contribution of linear component of electric charge

R s :

Switching process zone

\( {\overline{s}}_{ij}^E \) :

Elastic compliance constants measured at a constant electric field

S :

Width of specimen

S m :

Moment arm in DT test specimen

t :

Thickness of specimen

V :

Applied voltage

[Y]:

Hermitian matrix

∆:

(Load-point) mechanical displacement

Δ l :

Linear component of mechanical displacement

Δ F l :

Pure mechanical contribution of linear component of mechanical displacement

Δ P l :

Piezoelectric contribution of linear component of mechanical displacement

ε ij :

Strain components

θ :

Total angle of twist of torsion bar

κ :

Constant of Irwin matrix

κ ε ij or \( {\overline{\kappa}}_{ij}^{\varepsilon } \) :

Dielectric permittivities under constant strain

\( {\overline{\kappa}}_{ij}^{\sigma } \) :

Dielectric permittivities under constant stress

μ :

Shear modulus of isotropic specimen

Π :

Potential energy

Π l :

Linear part of the potential energy

σ ij :

Stress components

τ = 2 t/ S :

Thickness ratio

ψ(τ):

Finite beam thickness correction factor

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Acknowledgments

This work was supported by the Japan Society for the Promotion of Science (JSPS-P06118), expressed high gratitude for Prof. Watanabe Katsuhiko’s whole attention and cooperation.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Koje College, 91 Majeon 1gil, Geoje, Gyeongnam, 656-701, Republic of Korea

    D.-C. Shin

  2. School of Nanomech. Engr., Pusan National University, 1268-50, Samrangjin-ro, Miryang, Gyeongnam, 627-706, Republic of Korea

    T.-G. Kim

  3. Department of Aviation Maintenance, Daegu Technical University, Bon-dong, Dalseo-gu, Daegu, 704-721, Republic of Korea

    D.-W. Kim

Authors
  1. D.-C. Shin
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Corresponding author

Correspondence to D.-W. Kim.

Appendix A: Irwin Matrix

Appendix A: Irwin Matrix

The eigenvalues obtained by solution of problem in x 1 − x 2 plane for C-82 piezoelectric ceramic with poling direction of positive x 2-axis are

$$ \left\{p\right\}=\left\{{p}_1,\kern0.5em {p}_2,\kern0.5em {p}_3,\kern0.5em {p}_4\right\}=\left\{0.794797\mathrm{i},\kern0.5em 0.97323\mathrm{i},\kern0.5em 1.07225\mathrm{i},\kern0.5em 1.63569\mathrm{i}\right\} $$
(31)

where \( \mathrm{i}=\sqrt{-1} \).

When a crack was existed on piezoelectric materials, a Hermitian matrix, [Y], and a Irwin matrix, [H], can be defined as shown below [25]:

$$ \left[Y\right]=\left[\begin{array}{cccc}\hfill 0.0231538\hfill & \hfill 0.0107805\mathrm{i}\hfill & \hfill 0\hfill & \hfill -0.0122096\mathrm{i}\hfill \\ {}\hfill \hbox{-} 0.0107805\mathrm{i}\hfill & \hfill 0.0210542\hfill & \hfill 0\hfill & \hfill 0.0175581\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0.0459781\hfill & \hfill 0\hfill \\ {}\hfill 0.0122096\mathrm{i}\hfill & \hfill 0.0175581\hfill & \hfill 0\hfill & \hfill \hbox{-} 0.0362154\hfill \end{array}\right] $$
(32)
$$ \left[H\right]=\left[\begin{array}{cccc}\hfill 0.0463076\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0.0421084\hfill & \hfill 0\hfill & \hfill 0.0351163\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0.0919562\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0.0351163\hfill & \hfill 0\hfill & \hfill \hbox{-} 0.0724307\hfill \end{array}\right] $$
(33)

By Sosa and Pak [36], the matrix, [H], has following structure:

$$ \left[H\right]=\left[\begin{array}{cccc}\hfill 2/{c}_L\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 2/{c}_T\hfill & \hfill 0\hfill & \hfill 2/e\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 2/{c}_A\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 2/e\hfill & \hfill 0\hfill & \hfill -2/\kappa \hfill \end{array}\right] $$
(34)

Therefore, parameters, c L , c T , c A , e, and κ can be numerically calculated as shown in Tables 1 and 3.

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Shin, DC., Kim, TG. & Kim, DW. Electromechanical Linear Contribution at Fracture of Crack on Pre-notched Piezoelectric Ceramics Under Double Torsion. Exp Mech 55, 1729–1744 (2015). https://doi.org/10.1007/s11340-015-0081-6

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