The asymptotic homogenization method (AHM) yields a two-scale procedure to obtain the effective properties of a composite material containing a periodic distribution of unidirectional circular cylindrical holes in a linear transversely isotropic piezoelectric matrix. The matrix material belongs to the symmetry crystal class 622. The holes are centered in a periodic array of cells of square cross sections and the periodicity is the same in two perpendicular directions. The composite state is antiplane shear piezoelectric, that is, a coupled state of out-of-plane shear deformation and in-plane electric field. Local problems that arise from the two-scale analysis using the AHM are solved by means of a complex variable method. For this, the solutions are expanded in power series of Weierstrass elliptic functions, which contain coefficients that are determined from the solutions of infinite systems of linear algebraic equations. Truncating the infinite systems up to a finite, but otherwise arbitrary, order of approximation, we obtain analytical formulas for effective elastic, piezoelectric, and dielectric properties, which depend on both the volume fraction of the holes and an electromechanical coupling factor of the matrix. Numerical results obtained from these formulas are compared with results obtained by the Mori–Tanaka approach. The results could be useful in bone mechanics.

References

References
1.
Fukada
,
E.
, and
Yasuda
,
I.
,
1957
, “
On the Piezoelectric Effect of Bone
,”
J. Phys. Soc. Jpn.
,
12
, pp.
1158
1162
.10.1143/JPSJ.12.1158
2.
Nye
,
J. F.
,
1985
,
Physical Properties of Crystals: Their Representations by Tensors and Matrices
,
Oxford University Press
,
Oxford
, UK.
3.
Lang
,
S. B.
,
1970
, “
Ultrasonic Method for Measuring Elastic Coefficients of Bone and Results on Fresh and Dried Bovine Bones
,”
IEEE Trans. Biomed. Eng.
,
17
(
2
), pp.
101
105
.10.1109/TBME.1970.4502706
4.
Katz
,
J. L.
,
1980
, “
Anisotropy of Young's Modulus of Bone
,”
Nature
,
283
, p.
106
107
.10.1038/283106a0
5.
Hollister
,
S. J.
,
Fyhrie
,
D. P.
,
Jepsen
,
K. J.
, and
Goldstein
,
S. A.
,
1989
, “
Analysis of Trabecular Bone Micro-Mechanics Using Homogenization Theory With Comparison to Experimental Results
,”
J. Biomech.
,
22
, p.
1025
.10.1016/0021-9290(89)90284-4
6.
Fyhrie
,
D. P.
,
Jepsen
,
K. J.
,
Hollister
,
S. J.
, and
Goldstein
,
S. A.
,
1989
, “
Predicting Trabecular Bone Strength and Micro-Strain Using Homogenization Theory
,”
J. Biomech.
,
22
, p.
1014
.10.1016/0021-9290(89)90251-0
7.
Van Buskirk
,
W. C.
,
Cowin
,
S. C.
, and
Ward
,
R. N.
,
1981
, “
Ultrasonic Measurement of Orthotropic Elastic Constants of Bovine Femoral Bone
,”
ASME J. Biomech. Eng.
,
103
(
2
), pp.
67
72
.10.1115/1.3138262
8.
Grimal
,
Q.
,
Raum
,
K.
,
Gerisch
,
A.
, and
Laugier
,
P.
,
2009
, “
About the Determination of the Representative Volume Element Size in Compact Bone
,” 19o Congrès Français de Mécanique.
9.
Martin
,
R. B.
, and
Burr
,
D. B.
,
1989
,
Structure, Function and Adaptation of Compact Bone
,
Raven Press
,
New York
.
10.
Bloebaum
,
R. D.
, and
Isaacson
,
B. M.
,
2010
, “
Bone Bioelectricity: What Have We Learned in the Past 160 Years?
,”
J. Biomed. Mater. Res.
,
95A
(
4
), pp.
1270
1279
.10.1002/jbm.a.32905
11.
Minary-Jolandan
,
M.
, and
Min-Feng
,
Y.
,
2010
, “
Shear Piezoelectricity in Bone at the Nanoscale
,”
Appl. Phys. Lett.
,
97
(
15
), p.
153127
.10.1063/1.3503965
12.
Duarte
,
L. R.
,
1983
, “
The Stimulation of Bone Growth by Ultrasound
,”
Arch. Orthop. Trauma Surg.
,
101
, pp.
153
159
.10.1007/BF00436764
13.
Behari
,
J.
,
2009
,
Front Matter in Biophysical Bone Behavior: Principles and Applications
,
John Wiley
,
Chichester, UK
.
14.
The Institute of Radio Engineers
,
1958
, “
IRE Standards on Piezoelectric Crystals: Determination of the Elastic, Piezoelectric, and Dielectric Constants—The Electromechanical Coupling Factor
,”
Proc. IRE
,
46
(
4
), pp.
764
778
.10.1109/JRPROC.1958.286778
15.
Ikeda
,
T.
,
1990
,
Fundamentals of Piezoelectricity
,
Oxford University Press
,
Oxford
, UK.
16.
Bravo-Castillero
,
J.
,
Guinovart-Diaz
,
R.
,
Sabina
,
F. J.
, and
Rodríguez-Ramos
,
R.
,
2001
, “
Closed-Form Expressions for the Effective Coefficients of a Fiber-Reinforced Composite With Transversely Isotropic Constituents—II. Piezoelectric and Square Symmetry
,”
Mech. Mater.
,
33
, pp.
237
248
.10.1016/S0167-6636(00)00060-0
17.
Bravo-Castillero
,
J.
,
Rodríguez-Ramos
,
R.
,
Guinovart-Díaz
,
R.
,
Sabina
,
F. J.
,
Aguiar
,
A. R.
,
Silva
,
U. P.
, and
Gómez-Muñoz
,
J. L.
,
2009
, “
Analytical Formulas for Electromechanical Effective Properties of 3-1 Longitudinally Porous Piezoelectric Materials
,”
Acta Mater.
,
57
, pp.
795
803
.10.1016/j.actamat.2008.10.015
18.
Bravo-Castillero
,
J.
,
Guinovart-Díaz
,
R.
,
Rodríguez-Ramos
,
R.
,
Mechkour
,
H.
,
Brenner
,
R.
,
Camacho-Montes
,
H.
, and
Sabina
,
F. J.
,
2011
, “
Universal Relations and Effective Coefficients of Magneto-Electro-Elastic Perforated Structures
,”
Q. J. Mech. Appl. Math.
,
65
, pp.
61
85
.10.1093/qjmam/hbr020
19.
Lopez-Lopez
,
E.
,
Sabina
,
F. J.
,
Bravo-Castillero
,
J.
,
Guinovart-Diaz
,
R.
, and
Rodríguez-Ramos
,
R.
,
2005
, “
Overall Electromechanical Properties of a Binary Composite With 622 Symmetry Constituents. Antiplane Shear Piezoelectric State
,”
Int. J. Solids Struct.
,
42
(
21–22
), pp.
5765
5777
.10.1016/j.ijsolstr.2005.03.013
20.
Milgrom
,
M.
, and
Shtrikman
,
S.
,
1989
, “
Linear Response of Two-Phase Composites With Cross-Moduli: Exact Universal Relations
,”
Phys. Rev. A
,
40
, p.
1568
1575
.10.1103/PhysRevA.40.1568
21.
Mori
,
T.
, and
Tanaka
,
K.
,
1973
, “
Average Stress in Matrix and Average Elastic Energy of Materials With Misfitting Inclusions
,”
Acta Metall.
,
21
, p.
571
574
.10.1016/0001-6160(73)90064-3
22.
Armitage
,
J. V.
, and
Eberlein
,
W. F.
,
2006
,
Elliptic Functions
,
Cambridge University Press
,
Cambridge, UK
.
23.
Grigolyuk
,
E. I.
, and
Fil'shtinskii
,
L. A.
,
1970
,
Perforated Plates and Shells
,
Nauka
,
Moscow, in Russian
.
24.
Pobedrya
,
B. E.
,
1984
,
Mechanics of Composite Materials
,
Izd-vo MGU
,
Moscow, in Russian
.
25.
Markushevich
,
A.
,
1970
,
Teoría de las Funciones Analíticas
,
Mir
,
Moscow, in Spanish
.
26.
Yingzhen
,
L.
,
Minggen
,
C.
, and
Yi
,
Z.
,
2005
, “
Representation of the Exact Solution for Infinite System of Linear Equations
,”
Appl. Math. Comput.
,
168
(
1
), pp.
636
650
.10.1016/j.amc.2004.09.025
27.
Benveniste
,
Y.
,
1987
, “
A New Approach to the Application of Mori–Tanaka's Theory in Composite Materials
,”
Mech. Mater.
,
6
, p.
147
157
.10.1016/0167-6636(87)90005-6
28.
Dunn
,
M. L.
, and
Taya
,
M.
,
1993
, “
Micromechanics Predictions of the Effective Electroelastic Moduli of Piezoelectric Composites
,”
Int. J. Solids Struct.
,
30
, pp.
161
175
.10.1016/0020-7683(93)90058-F
29.
Dunn
,
M. L.
, and
Taya
,
M.
,
1993
, “
Electromechanical Properties of Porous Piezoelectric Ceramics
,”
J. Am. Ceram. Soc.
,
76
, p.
1697
1706
.10.1111/j.1151-2916.1993.tb06637.x
30.
Eshelby
,
J. D.
,
1957
, “
The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems
,”
Proc. R. Soc. London Ser. A
,
241
, pp.
376
396
.10.1098/rspa.1957.0133
31.
Zohdi
,
T. I.
, and
Wriggers
,
P.
,
2008
,
An Introduction to Computational Micromechanics
,
2nd ed.
(Lecture Notes in Applied and Computational Mechanics, Vol.
20
),
Springer
,
Berlin
.
32.
Kar-Gupta
,
R.
, and
Venkatesh
,
T. A.
,
2006
, “
Electromechanical Response of Porous Piezoelectric Materials
,”
Acta Mater.
,
54
, pp.
4063
4078
.10.1016/j.actamat.2006.04.037
33.
Mikata
,
Y.
,
2000
, “
Determination of Piezoelectric Eshelby Tensor in Transverselyisotropic Piezoelectric Solids
,”
Int. J. Eng. Sci.
,
38
, pp.
605
641
.10.1016/S0020-7225(99)00050-6
34.
Berlincourt
,
D. A.
,
Curran
,
D. R.
, and
Jaffe
,
H.
,
1964
,
Piezoelectric and Piezomagnetic Materials and Their Function in Transducers
,
2nd ed.
(Physical Acoustics, Principles and Methods, Vol.
1
, Part a3),
Academic Press
,
New York
, pp.
169
270
.
35.
Gundjian
,
A. A.
, and
Chen
,
H. L.
,
1974
, “
Standardization and Interpretation of Electromechanical Properties of Bone
,”
IEEE Trans. Biomed. Eng.
,
21
(
3
), pp.
177
182
.10.1109/TBME.1974.324380
36.
Parnell
,
W. J.
, and
Grimal
,
Q.
,
2009
, “
The Influence of Mesoscale Porosity on Cortical Bone Anisotropy. Investigations Via Asymptotic Homogenization
,”
J. R. Soc. Interface
,
6
, pp.
97
109
.10.1098/rsif.2008.0255
You do not currently have access to this content.