30
2 Literature Review
154. J.S. Yang, Equations for the flexural motion of elastic plates with partially electroded piezoelectric actuators. Smart Mater. Struct. 6, 485–490 (1997)
155. J.S. Yang, Equations for thick elastic plates with partially electroded piezoelectric actuators
and higher order electric fields. Smart Mater. Struct. 8, 73–82 (1999)
156. S.V. Gopinathan, V.V. Varadan, V.K. Varadan, A review and critique of theories for piezoelectric laminates. Smart Mater. Struct. 9, 24–48 (2000)
157. V. Cotoni, P. Masson, F. Côté, A finite element for piezoelectric multilayered plates: combined
higher-order and piecewise linear c 0 formulation. J. Intell. Mater. Syst. Struct. 17, 155–166
(2006)
158. M. Kögl, M.A. Bucalem, Analysis of smart laminates using piezoelectric MITC plate and
shell elements. Comput. Struct. 83, 1153–1163 (2005)
159. D. Marinkovi´ c, H. Köppe, U. Gabbert, Degenerated shell element for geometrically nonlinear
analysis of thin-walled piezoelectric active structures. Smart Mater. Struct. 17, 1–10 (2008)
160. D. Marinkovi´ c, H. Köppe, U. Gabbert, Aspects of modeling piezoelectric active thin-walled
structures. J. Intell. Mater. Syst. Struct. 20, 1835–1844 (2009)
161. D.F. Nelson, Theory of nonlinear electroacoustics of dielectric, piezoelectric, and pyroelectric
crystals. J. Acoust. Soc. Am. 63(6), 1738–1748 (1978)
162. S.P. Joshi, Non-linear constitutive relations for piezoceramic materials. Smart Mater. Struct.
1, 80–83 (1992)
163. J.S. Yang, R.C. Batra, A second-order theory for piezoelectric materials. J. Acoust. Soc. Am.
97(1), 280–288 (1995)
164. M. Kamlah, U. Böhle, Finite element analysis of piezoceramic components taking into account
ferroelectric hysteresis behavior. Int. J. Solids Struct. 38, 605–633 (2001)
165. X. Zhou, A. Chattopadhyay, Nonlinear piezoelectric constitutive relationship and actuation for
piezoelectric laminates, in 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (Denver, Colorado, 22–25 April 2002), pp. AIAA–2002–1438
166. C.M. Landis, Non-linear constitutive modeling of ferroelectrics. Curr. Opin. Solid State Mater.
Sci. 8, 59–69 (2004)
167. M. Elhadrouz, T.B. Zineb, E. Patoor, Finite element analysis of a multilayer piezoelectric
actuator taking into account the ferroelectric and ferroelastic behaviors. Int. J. Eng. Sci. 44,
996–1006 (2006)
168. L. Ma, Y. Shen, J. Li, H. Zheng, T. Zou, Modeling hysteresis for piezoelectric actuators. J.
Intell. Mater. Syst. Struct. 27(10), 1404–1411 (2016)
169. S. Li, W. Cao, L.E. Cross, The extrinsic nature of nonlinear behavior observed in lead zirconate
titanate ferroelectric ceramic. J. Appl. Phys. 69(10), 7219–7224 (1991)
170. A.J. Masys, W. Ren, G. Yang, B.K. Mukherjee, Piezoelectric strain in lead zirconate titante
ceramics as a function of electric field, frequency, and dc bias. J. Appl. Phys. 94(2), 1155–1162
(2003)
171. S. Klinkel, A phenomenological constitutive model for ferroelastic and ferroelectric hysteresis
effects in ferroelectric ceramics. Int. J. Solids Struct. 43, 7197–7222 (2006)
172. K. Linnemann, S. Klinkel, W. Wagner, A constitutive model for magnetostrictive and piezoelectric materials. Int. J. Solids Struct. 46, 1149–1166 (2009)
173. P. Tan, L. Tong, A one-dimensional model for non-linear behaviour of piezoelectric composite
materials. Compos. Struct. 58(4), 551–561 (2002)
174. Q.M. Wang, Q. Zhang, B. Xu, R. Liu, L.E. Cross, Nonlinear piezoelectric behavior of ceramic
bending mode actuators under strong electric fields. J. Appl. Phys. 86(6), 3352–3360 (1999)
175. L.Q. Yao, J.G. Zhang, L. Lu, M.O. Lai, Nonlinear dynamic characteristics of piezoelectric
bending actuators under strong applied electric. J. Microelectromechanical Syst. 13(4), 645–
652 (2004)
176. D. Sun, L. Tong, D. Wang, An incremental algorithm for static shape control of smart structures
with nonlinear piezoelectric actuators. Int. J. Solids Struct. 41, 2277–2292 (2004)
177. Z.K. Kusculuoglu, T.J. Royston, Nonlinear modeling of composite plates with piezoceramic
layers using finite element analysis. J. Sound Vib. 315, 911–926 (2008)
2 Literature Review
154. J.S. Yang, Equations for the flexural motion of elastic plates with partially electroded piezoelectric actuators. Smart Mater. Struct. 6, 485–490 (1997)
155. J.S. Yang, Equations for thick elastic plates with partially electroded piezoelectric actuators
and higher order electric fields. Smart Mater. Struct. 8, 73–82 (1999)
156. S.V. Gopinathan, V.V. Varadan, V.K. Varadan, A review and critique of theories for piezoelectric laminates. Smart Mater. Struct. 9, 24–48 (2000)
157. V. Cotoni, P. Masson, F. Côté, A finite element for piezoelectric multilayered plates: combined
higher-order and piecewise linear c 0 formulation. J. Intell. Mater. Syst. Struct. 17, 155–166
(2006)
158. M. Kögl, M.A. Bucalem, Analysis of smart laminates using piezoelectric MITC plate and
shell elements. Comput. Struct. 83, 1153–1163 (2005)
159. D. Marinkovi´ c, H. Köppe, U. Gabbert, Degenerated shell element for geometrically nonlinear
analysis of thin-walled piezoelectric active structures. Smart Mater. Struct. 17, 1–10 (2008)
160. D. Marinkovi´ c, H. Köppe, U. Gabbert, Aspects of modeling piezoelectric active thin-walled
structures. J. Intell. Mater. Syst. Struct. 20, 1835–1844 (2009)
161. D.F. Nelson, Theory of nonlinear electroacoustics of dielectric, piezoelectric, and pyroelectric
crystals. J. Acoust. Soc. Am. 63(6), 1738–1748 (1978)
162. S.P. Joshi, Non-linear constitutive relations for piezoceramic materials. Smart Mater. Struct.
1, 80–83 (1992)
163. J.S. Yang, R.C. Batra, A second-order theory for piezoelectric materials. J. Acoust. Soc. Am.
97(1), 280–288 (1995)
164. M. Kamlah, U. Böhle, Finite element analysis of piezoceramic components taking into account
ferroelectric hysteresis behavior. Int. J. Solids Struct. 38, 605–633 (2001)
165. X. Zhou, A. Chattopadhyay, Nonlinear piezoelectric constitutive relationship and actuation for
piezoelectric laminates, in 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (Denver, Colorado, 22–25 April 2002), pp. AIAA–2002–1438
166. C.M. Landis, Non-linear constitutive modeling of ferroelectrics. Curr. Opin. Solid State Mater.
Sci. 8, 59–69 (2004)
167. M. Elhadrouz, T.B. Zineb, E. Patoor, Finite element analysis of a multilayer piezoelectric
actuator taking into account the ferroelectric and ferroelastic behaviors. Int. J. Eng. Sci. 44,
996–1006 (2006)
168. L. Ma, Y. Shen, J. Li, H. Zheng, T. Zou, Modeling hysteresis for piezoelectric actuators. J.
Intell. Mater. Syst. Struct. 27(10), 1404–1411 (2016)
169. S. Li, W. Cao, L.E. Cross, The extrinsic nature of nonlinear behavior observed in lead zirconate
titanate ferroelectric ceramic. J. Appl. Phys. 69(10), 7219–7224 (1991)
170. A.J. Masys, W. Ren, G. Yang, B.K. Mukherjee, Piezoelectric strain in lead zirconate titante
ceramics as a function of electric field, frequency, and dc bias. J. Appl. Phys. 94(2), 1155–1162
(2003)
171. S. Klinkel, A phenomenological constitutive model for ferroelastic and ferroelectric hysteresis
effects in ferroelectric ceramics. Int. J. Solids Struct. 43, 7197–7222 (2006)
172. K. Linnemann, S. Klinkel, W. Wagner, A constitutive model for magnetostrictive and piezoelectric materials. Int. J. Solids Struct. 46, 1149–1166 (2009)
173. P. Tan, L. Tong, A one-dimensional model for non-linear behaviour of piezoelectric composite
materials. Compos. Struct. 58(4), 551–561 (2002)
174. Q.M. Wang, Q. Zhang, B. Xu, R. Liu, L.E. Cross, Nonlinear piezoelectric behavior of ceramic
bending mode actuators under strong electric fields. J. Appl. Phys. 86(6), 3352–3360 (1999)
175. L.Q. Yao, J.G. Zhang, L. Lu, M.O. Lai, Nonlinear dynamic characteristics of piezoelectric
bending actuators under strong applied electric. J. Microelectromechanical Syst. 13(4), 645–
652 (2004)
176. D. Sun, L. Tong, D. Wang, An incremental algorithm for static shape control of smart structures
with nonlinear piezoelectric actuators. Int. J. Solids Struct. 41, 2277–2292 (2004)
177. Z.K. Kusculuoglu, T.J. Royston, Nonlinear modeling of composite plates with piezoceramic
layers using finite element analysis. J. Sound Vib. 315, 911–926 (2008)