Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-14T03:29:50.033Z Has data issue: false hasContentIssue false

Numerical Evaluation of Size Effect in Piezoelectric Micro-Beam with Linear Micromorphic Electroelastic Theory

Published online by Cambridge University Press:  22 May 2014

W.-Z. Cao
Affiliation:
Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China
X.-H. Yang*
Affiliation:
Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China
X.-B. Tian
Affiliation:
Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China
Get access

Abstract

The linear micromorphic electroelastic theory is proposed to solve bending problems of piezoelectric micro-beam in this paper. The basic governing equations with the boundary conditions are derived through the variational principle. Both the cantilever piezoelectric micro-beam subjected to a concentrated load at the free end and the simply supported micro-beam subjected to a distributed load are analyzed. It is found that the predictions from the micromorphic electroelastic theory are remarkably different from those from the classical theory when the micro-beam thickness is approximate or equal to the characteristic length scale parameter, but their difference is slight when the micro-beam thickness is much larger than the characteristic length scale parameter. As a result, it is concluded that the size effect is significant when the micro-beam thickness is comparable to the characteristic length scale parameter.

Type
Research Article
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Kong, X. Y. and Wang, Z. L., “Polar-Surface Dominated Zno Nanobelts and the Electrostatic Energy Induced Nanohelixes, Nanosprings, and Nanospi-rals,” Applied Physics Letters, 84, pp. 975977 (2004).Google Scholar
2.Jian, J. K., Zhang, Z. H., Sun, Y. P., Lei, M., Chen, X. L., Wang, T. M. and Wang, C., “Gan Nanorings: Another Example of Spontaneous Polarization-Induced Nanostructure,” Journal of Crystal Growth, 303, pp. 427432 (2007).Google Scholar
3.Majidi, C., Chen, Z., Srolovitz, D. J. and Haataja, M., “Spontaneous Bending of Piezoelectric Nanoribbons: Mechanics, Polarization, and Space Charge Coupling,” Journal of Mechanics and Physics of Solids, 58, pp. 7385 (2010).CrossRefGoogle Scholar
4.Multani, M. S. and Palkar, V. R., “Morphotropic Phase Boundary in the PZT System,” Materials Reviews B, 17, pp. 101104 (1982).Google Scholar
5.Mishima, T., Fujioka, H., Nagakari, S., Kamigake, K. and Nambu, S., “Lattice Image Observations of Na-noscale Ordered Regions in Pb(Mg1/3Nb2/3)O3,” Japan Journal of Applied Physics, 36, pp. 61416144 (1997).Google Scholar
6.Eringen, A. C. and Suhubi, E. S., “Nonlinear Theory of Simple Micro-Elastic Solids - I,” International Journal of Engineering Science, 2, pp. 189203 (1964).Google Scholar
7.Eringen, A. C. and Suhubi, E. S., “Nonlinear Theory of Simple Micro-Elastic Solids - II,” International Journal of Engineering Science, 2, pp. 389404 (1964).Google Scholar
8.Mindlin, R. D., “Micro-Structure in Linear Elasticity,” Archives Rational Mechanics and Analysis, 16, pp. 5178 (1964).CrossRefGoogle Scholar
9.Eringen, A. C., Microcontinuum Field Theories. I: Foundations and Solids, Springer, Berlin (1999).CrossRefGoogle Scholar
10.Iesan, D., “On the Micromorphic Thermoelasticity,” International Journal of Engineering Science, 40, pp. 549567 (2002).Google Scholar
11.Dillard, T., Forest, S. and Ienny, P., “Micromorphic Continuum Modelling of the Deformation and Fracture Behaviour of Nickel Foams,” European Journal of Mechanics A-Solid, 25, pp. 526549 (2006).CrossRefGoogle Scholar
12.Neff, P. and Forest, S., “A Geometrically Exact Micromorphic Model for Elastic Metallic Foams Accounting for Affine Microstructure. Modelling, Existence of Minimizers, Identification of Moduli and Computational Results,” Journal of Elasticity, 87, pp. 239276 (2007).Google Scholar
13.Forest, S. and Sievert, R., “Elastoviscoplastic Constitutive Frameworks for Generalized Continua,” Acta Mechanica, 160, pp. 71111 (2003).CrossRefGoogle Scholar
14.Forest, S. and Sievert, R., “Nonlinear Microstrain Theories,” International Journal of Solids and Structures, 43, pp. 72247245 (2006).CrossRefGoogle Scholar
15.Forest, S., “Micromorphic Approach for Gradient Elasticity, Viscoplasticity, and Damage,” Journal of Engineering Mechanics, ASCE, 135, pp. 117131 (2009).Google Scholar
16.Cordero, N. M., Gaubert, A., Forest, S., Busso, E. P., Gallerneau, F. and Kruch, S., “Size Effects in Generalised Continuum Crystal Plasticity for Two-Phase Laminates,” Journal of Mechanics and Physics of Solids, 58, pp. 19631994 (2010).Google Scholar
17.Lazar, M. and Maugin, G., “On Microcontinuum Field Theories: The Eshelby Stress Tensor and Incompatibility Conditions,” Philosophical Magazine, 87. pp. 38533870 (2007).Google Scholar
18.Eringen, A. C., “Continuum Theory of Micromorphic Electromagnetic Thermoelastic Solids,” International Journal of Engineering Science, 41, pp. 653665 (2003).Google Scholar
19.Lee, J. D., Chen, Y. P. and Eskandarian, A., “A Mi-cromorphic Electromagnetic Theory,” International Journal of Solids and Structures, 41, pp. 20992110 (2004).Google Scholar
20.Lee, J. D. and Chen, Y. P., “Electromagnetic Wave Propagation in Micromorphic Elastic Solids,” International Journal of Engineering Science, 42, pp. 841848 (2004).Google Scholar
21.Eringen, A. C., “Micromorphic Electromagnetic Theory and Waves,” Foundations of Physics, 36, pp. 902919 (2006).Google Scholar
22.Cao, W. Z. and Yang, X. H., “Families of Generalized Variational Principles for Linear Micromorphic Electroelasticity and Electroelastodynamics,” International Journal of Nonlinear Science and Numerical, 11, pp. 419427 (2010).Google Scholar
23.Cao, W. Z., Yang, X. H. and Tian, X. B., “Anti-Plane Problems of Piezoelectric Material with a Micro-Void or Micro-Inclusion Based on Micromorphic Electroelastic Theory,” International Journal of Solids and Structures, 49, pp. 31853200 (2012).Google Scholar
24.Mindlin, R. D. and Eshel, N. N., “On First Strain Gradient Theories in Linear Elasticity,” International Journal of Solids and Structures, 4, pp. 109124 (1968).CrossRefGoogle Scholar
25.Jackson, J. D., Classical Electrodynamics, 3rd Edition, Wiley, New York (1998)Google Scholar
26.Yang, J. S. and Fang, H. Y., “Analysis of a Rotating Elastic Beam with Piezoelectric Films as an Angular Rate Sensor,” IEEE Transactions on Ultrasonics, Ferroelectrics, 49, pp. 798804 (2002).Google Scholar
27.Lazopoulos, K. A. and Lazopoulos, A. K., “Bending and Buckling of Thin Strain Gradient Elastic Beams,” European Journal of Mechanics A-Solid, 29, pp. 837843 (2010).CrossRefGoogle Scholar
28.Yang, X. H., Chen, C. Y., Hu, Y. T. and Wang, C., “Damage Analysis and Fracture Criteria for Piezoelectric Ceramics,” International Journal of Nonlinear Mechanics, 40, pp. 12041213 (2005).Google Scholar
29.Yang, X. H., Chen, C. Y. and Hu, Y. T., “Analysis of Damage Near a Conducting Crack in a Piezoelectric Ceramic,” Acta Mechanica Solida Sinica, 16, pp. 147154 (2003).Google Scholar
30.Li, L. and Xie, S. S., “Finite Element Method for Linear Micropolar Elasticity and Numerical Study of Some Scale Effects Phenomena in MEMS,” International Journal of Mechanical Sciences, 46, pp. 15711587 (2004).Google Scholar
31.Wang, G. F., Yu, S. W. and Feng, X. Q., “A Piezoelectric Constitutive Theory with Rotation Gradient Effects,” European Journal of Mechanics A-Solid, 23, pp. 455466 (2004).Google Scholar
32.Chen, Z., Majidi, C., Srolovitz, D. J. and Haataja, M., “Tunable Helical Ribbons,” Applied Physics Letters, 98, p. 011906 (2011).Google Scholar
33.Chen, Z., Guo, Q., Majidi, C., Chen, W., Srolovitz, D. J. and Haataja, M., “Nonlinear Geometric Effects in Mechanical Bistable Morphing Structures,” Physical Review Letters, 109, p. 114302 (2012).CrossRefGoogle ScholarPubMed
34.Fleck, N. A., Muller, G. M., Ashby, M. F. and Hutchinson, J. W., “Strain Gradient Plasticity: Theory and Experiment,” Acta Metallurgica and Mate-rialia, 42, pp. 475487 (1994).Google Scholar
35.Xia, Z. C. and Hutchinson, J. W., “Crack Tip fields in Strain Gradient Plasticity,” Journal of Mechanics and Physics of Solids, 44, pp. 16211648 (1996).Google Scholar
36.Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J. and Tong, P., “Experiments and Theory in Strain Gradient Elasticity,” Journal of Mechanics and Physics of Solids, 51, pp. 14771508 (2003).Google Scholar
37.Nakamura, S. and Lakes, R., “Finite Element Analysis of Saint-Venant End Effects in Micropolar Elastic Solids,” Engineering Computation, 12, pp. 571587 (1995).Google Scholar