Planar four node piezoelectric elements with drilling degrees of freedom

International Journal for Numerical Methods in Engineering

Volumn 65, Issue 11, 2006, Pages 1802-1830

  Long, C S ^a   Loveday, P W ^b   Groenwold, Albert A ^a  

^a UNIVERSITY OF PRETORIA   (South Africa)

^b Future production Chemicals Cluster   (South Africa)

Author keywords

Drilling degrees of freedom; Finite element; Mixed variational formulation; Piezoelectricity

Indexed keywords

DEGREES OF FREEDOM (MECHANICS); DRILLING; FINITE ELEMENT METHOD; MATHEMATICAL MODELS; STRESSES; TENSORS;

DRILLING DEGREES OF FREEDOM; ELECTRIC FLUX DENSITY; MIXED VARIATIONAL FORMULATION; PIEZOELECTRIC ELEMENTS; PLANAR NODE; STRESS TENSOR;

PIEZOELECTRICITY;

DEGREES OF FREEDOM (MECHANICS); DRILLING; FINITE ELEMENT METHOD; MATHEMATICAL MODELS; PIEZOELECTRICITY; STRESSES; TENSORS;

EID: 33644822945     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.1524     Document Type: Article

References (48)

1. Heyliger P. A note on the static behaviour of simply-supported laminated piezoelectric cylinders. International Journal of Solids and Structures 1997; 34:3781-3794.
2. Kapuria S, Sengupta S, Dumir PC. Three-dimensional solution for simply-supported piezoelectric cylindrical shell for axisymmetric load. Computer Methods in Applied Mechanics and Engineering 1997; 140:139-155.
3. Allik H, Hughes TJR. Finite element method for piezoelectric vibration. International Journal for Numerical Methods in Engineering 1970; 2:151-157.
4. Benjeddou A. Advances in piezoelectric finite element modeling of adaptive structural elements: a survey. Computers and Structures 2000; 76:347-363.
5. Yang JS. Mixed variational principles for piezoelectric elasticity. In Developments in Theoretical and Applied Mechanics, Antar B, Engels R, Prinaris AA, Moulden TH, Zanaboni O (eds), vol. XVI, University of Tennessee Space Institute: Tennessee, 1992; II.1.31-38.
6. EerNisse EP. Variational method for electroelastic vibration analysis. IEEE Transactions on Sonics and Ultrasonics 1967; SU-14:153-160.
7. Holland R, EerNisse EP. Variational evaluation of admittances of multielectroded three-dimensional piezoelectric structures. IEEE Transactions on Sonics and Ultrasonics 1968; SU-15:119-132.
8. Cannarozzi AA, Ubertini F. Some hybrid variational methods for linear electroelasticity problems. International Journal of Solids and Structures 2001; 38:2573-2596.
9. Sze KY, Yang X-M, Yao L-Q. Stabilized plane and axisymmetric piezoelectric finite element models. Finite Elements in Analysis and Design 2004; 40:1105-1122.
10. Wu CC, Sze KY, Huang YQ. Numerical solutions on fracture of piezoelectric materials by hybrid element. International Journal of Solids and Structures 2001; 38:4315-4329.
11. Sze KY, Yao LQ, Yi S. A hybrid stress ANS solid-shell element and its generalization for smart structure modelling. Part II - smart structure modelling. International Journal for Numerical Methods in Engineering 2000; 48:565-582.
12. Sze KY, Yao LQ. Modelling smart structures with segmented piezoelectric sensors and actuators. Journal of Sound and Vibration 2000; 235:495-520.
13. Sze KY, Pan YS. Hybrid finite element models for piezoelectric materials. Journal of Sound and Vibration 1999; 226:519-547.
14. Allman DJ. A compatible triangular element including vertex rotations for plane elasticity analysis. Computers and Structures 1984; 19:1-8.
15. Bergan PG, Felippa CA. A triangular membrane element with rotational degrees of freedom. Computer Methods in Applied Mechanics and Engineering 1985; 50:25-69.
16. Frey F. Shell finite elements with six degrees of freedom per node. ASME Winter Annual Meeting, San Francisco, 1989.
17. Jetteur P. A shallow shell element with in-plane rotational degrees of freedom. IREM Internal Report 86/3, Ecole Polytechnique Fédérale de Lausanne, 1986.
18. Jaamei S. 'Jet' thin shell finite element with drilling rotations. IREM Internal Report 88/7, Ecole Polytechnique Fédérale de Lausanne, 1988.
19. Jetteur P. Improvement of the quadrilateral 'JET' shell element for a particular class of shell problems. IREM Internal Report 87/1, Ecole Polytechnique Fédérale de Lausanne, 1987.
20. Taylor RL. Finite element analysis of linear shell problems. In The Mathematics of Finite Elements and Applications VI, MAFELAP, Whiteman JR (ed). Academic Press Limited: London, 1987; 191-203.
21. Taylor RL, Simo JC. Bending and membrane elements for analysis of thick and thin shells. In Proceedings of the NUMETA 85 Conference, Pande GN, Middleton J (eds), Swansea, 7-11 January 1985.
22. Simo JC, Fox DD, Rifai MS. On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects. Computer Methods in Applied Mechanics and Engineering 1989; 73:53-92.
23. Hughes TJR, Brezzi F. On drilling degrees of freedom. Computer Methods in Applied Mechanics and Engineering 1989; 72:105-121.
24. Reissner E. A note on variational principles in elasticity. International Journal of Solids and Structures 1965; 1:93-95.
25. Hughes TJR, Brezzi F, Masud A, Harari I. Finite element with drilling degree of freedom: theory and numerical evaluation. In Proceedings of the Fifth International Symposium on Numerical Methods in Engineering, Gruber R, Periaux J, Shaw RP (eds), vol. 1. Springer: Berlin, 1989; 3-17.
26. Ibrahimbegovic A, Taylor RL, Wilson EL. A robust quadrilateral membrane finite element with drilling degrees of freedom. International Journal for Numerical Methods in Engineering 1990; 30:445-457.
27. Ibrahimbegovic A, Wilson EL. A unified formulation for triangular and quadrilateral flat shell finite elements with six nodal degrees of freedom. Communications in Applied Numerical Methods 1991; 7:1-9.
28. Sze KY, Wanji C, Cheung YK. An efficient quadrilateral plane element with drilling degrees of freedom using orthogonal stress modes. Computers and Structures 1992; 42:695-705.
29. Sze KY, Ghali A. Hybrid plane quadrilateral element with corner rotations. ASCE Journal of Structural Engineering 1993; 119:2552-2572.
30. Groenwold AA, Stander N. An efficient 4-node 24 d.o.f. thick shell finite element with 5-point quadrature. Engineering Computations 1995; 12:723-748.
31. Groenwold AA, Stander N. A 24 d.o.f. 4-node flat shell finite element for general unsymmetric orthotropic layered composites. Engineering Computations 1998; 15:518-543.
32. Dovey HH. Extension of three dimensional analysis to shell structures using the finite element idealization. Ph.D. Thesis, University of California, Berkeley, 1974, Report No. UC SESM 74-2.
33. Geyer S, Groenwold AA. Two hybrid stress membrane finite element families with drilling rotations. International Journal for Numerical Methods in Engineering 2002; 53:583-601.
34. Long CS, Loveday PW, Groenwold AA. Design of a piezoelectric mirror scanning device using topology optimization. Proceedings of the Sixth World Congress on Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil, May 2005, Paper No. 4031.
35. Pimpinelli G. An assumed strain quadrilateral element with drilling degrees of freedom. Finite Elements in Analysis and Design 2004; 41:267-283.
36. Ikeda T. Fundamentals of Piezoelectricity. Oxford University Press: Oxford, 1996.
37. Long CS. Refined piezoelectric and electroelastic finite elements with applications in topology optimization. Technical Report 2005/T/01, Structural Optimization Research Group, Department of Mechanical Engineering, University of Pretoria, 2005. (Draft Ph.D. Thesis).
38. Simo JC, Rifai MS. A class of mixed assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering 1990; 29:1595-1638.
39. Ibrahimbegovic A, Frey F. Geometrically non-linear method of incompatible modes in application to finite elasticity with independent rotations. International Journal for Numerical Methods in Engineering 1993; 36:4185-4200.
40. Liu M. Finite element analysis for cracked piezoelectric material. Master's Thesis, Department of Modern Mechanics, University of Science and Technology of China, Hefei, 1998.
41. Di S, Ramm E. On alternative hybrid stress 2D and 3D elements. Engineering Computations 1994; 11:49-68.
42. Hughes TJR, Masud A, Harari I. Numerical assessment of some membrane elements with drilling degrees of freedom. Computers and Structures 1995; 55:297-314.
43. Long CS, Geyer S, Groenwold AA. A numerical study of the effect of penalty parameters for membrane elements with independent rotation fields and penalized equilibrium, 2005, submitted.
44. Geyer S, Groenwold AA. On reduced integration and locking of flat shell finite elements with drilling rotations. Communications in Numerical Methods in Engineering 2003; 19:85-97.
45. Hughes TJR, Masud A, Harari I. Dynamic analysis and drilling degrees of freedom. International Journal for Numerical Methods in Engineering 1995; 38:3193-3210.
46. Tzou HS. Piezoelectric Shells: Distributed Sensing and Control of Continua. Kluwer Academic Publishers: Netherlands, 1993.
47. Franco Correia VM, Aguiar Gomes MA, Suleman A, Mota Soares CM, Mota Soares CA. Modelling and design of adaptive composite structures. Computer Methods in Applied Mechanics and Engineering 2000; 185:325-346.
48. MacNeal RH, Harder RL. A proposed standard set of problems to test finite element accuracy. Finite Elements in Analysis and Design 1985; 1:3-20.