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AN EXISTENCE AND UNIQUENESS RESULT FOR AN ELASTO-PIEZOELECTRIC PROBLEM WITH DAMAGE

Departamento de Matemática Aplicada, Universidade de Santiago de Compostela Facultade de Matemáticas, Campus Sur s/n, 15782 Santiago de Compostela, Spain

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and

KENNETH L. KUTTLER

Department of Mathematics, Brigham Young University, Provo, UT 84602, USA

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https://doi.org/10.1142/S0218202509003346Cited by:5 (Source: Crossref)

Abstract

The aim of this paper is to study the damage evolution in an elasto-piezoelectric body. The effect of the damage, due to internal tension or compression and caused by the opening and growth of micro-cracks and micro-cavities, and the piezoelectric effects are included into the model. The variational formulation leads to a coupled system of evolutionary equations. An existence and uniqueness result is then proved by using the theory of maximal monotone operators, the Schauder fixed-point theorem, and a comparison result.

Keywords:

AMSC: 74D10, 74R05, 74F99, 35A05, 35A15, 35J20

References

T. A. Angelov, Mech. Res. Comm. 26, 281 (1999), DOI: 10.1016/S0093-6413(99)00025-7. Crossref, Web of Science, Google Scholar
C. Baiocchi, Ann. Mat. Pura Appl. 76, 233 (1967). Crossref, Google Scholar
H. Brezis , Opérateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espaces de Hilbert , Math. Studies 5 ( North-Holland , 1973 ) . Google Scholar
R. C. Batra and J. S. Yang, J. Elasticity 38, 209 (1995), DOI: 10.1007/BF00042498. Crossref, Web of Science, Google Scholar
G. Duvaut and J. L. Lions , Inequalities in Mechanics and Physics ( Springer-Verlag , 1976 ) . Crossref, Google Scholar
M. Frémond and B. Nedjar, Nucl. Engrg. Design 156, 323 (1995). Crossref, Web of Science, Google Scholar
M. Frémond and B. Nedjar, Internat. J. Solids Structures 33, 1083 (1996). Crossref, Web of Science, Google Scholar
M. Frémond , Non-Smooth Thermomechanics ( Springer , 2002 ) . Crossref, Google Scholar
T. Ideka , Fundamentals of Piezoelectricity ( Oxford Univ. Press , 1990 ) . Google Scholar
K. L. Kuttler, Electron. J. Diff. Eqns. 2005, 1 (2005). Google Scholar
K. L. Kuttler and M. Shillor, Nonlinear Anal. Real World Appl. 7, 674 (2006), DOI: 10.1016/j.nonrwa.2005.03.026. Crossref, Web of Science, Google Scholar
J. Lemaitre and R. Desmorat , Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures ( Springer-Verlag , 2005 ) . Google Scholar
R. Liebe, P. Steinmann and A. Benallal, Comput. Meth. Appl. Mech. Engrg. 190, 6555 (2001), DOI: 10.1016/S0045-7825(01)00250-X. Crossref, Web of Science, Google Scholar
J. L. Lions , Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires ( Dunod , 1969 ) . Google Scholar
C. Miehe, Eur. J. Mech. A Solids 14, 697 (1995). Web of Science, Google Scholar
R. D. Mindlin, Internat. J. Solids Structures 4, 1197 (1969). Google Scholar
R. D. Mindlin, J. Elasticity 4, 217 (1972). Google Scholar
B. Nedjar, Internat. J. Solids Structures 38, 5421 (2001), DOI: 10.1016/S0020-7683(00)00358-9. Crossref, Web of Science, Google Scholar
R. H. J. Peerlingset al., Internat. J. Numer. Meth. Engrg. 39, 3391 (1996). Crossref, Web of Science, Google Scholar
J. Simon, Ann. Mat. Pura. Appl. 146, 65 (1987), DOI: 10.1007/BF01762360. Crossref, Web of Science, Google Scholar
M. Sofonea and R. Tarraf, Appl. Anal. 86, 503 (2007), DOI: 10.1080/00036810701286304. Crossref, Web of Science, Google Scholar
R. A. Toupin, J. Rational Mech. Anal. 5, 849 (1956). Web of Science, Google Scholar
N. Turbé and G. A. Maugin, Math. Meth. Appl. Sci. 14, 403 (1991). Crossref, Web of Science, Google Scholar
Y. Xinhuaet al., Acta Mech. Solida Sin. 18, 21 (2005). Web of Science, Google Scholar