Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T18:12:06.564Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Numerical Modeling ofDeformations of Pressurized Martensitic Thin Films

Published online by Cambridge University Press:  15 April 2002

Pavel Bělík ,
Timothy Brule  and
Mitchell Luskin
Pavel Bělík
Affiliation:
School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455 USA. ([email protected]); ([email protected]); ([email protected])
Timothy Brule
Affiliation:
School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455 USA. ([email protected]); ([email protected]); ([email protected])
Mitchell Luskin
Affiliation:
School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455 USA. ([email protected]); ([email protected]); ([email protected])
Get access

Abstract

We propose, analyze, and compare several numerical methods for the computation of the deformation of a pressurized martensitic thin film. Numerical results have been obtained for the hysteresis of the deformation as the film transforms reversibly from austenite to martensite.

Keywords

Thin filmfinite elementmartensitic transformationactive materials.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 3 , May 2001 , pp. 525 - 548
Copyright
© EDP Sciences, SMAI, 2001

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

R. Adams, Sobolev spaces. Academic Press, New York (1975).
Argyris, J.H., Fried, I. and Scharpf, D.W., The TUBA family of plate elements for the matrix displacement method. Aero. J. Roy. Aero. Soc. 72 (1968) 701-709.
N.W. Ashcroft and N.D. Mermin, Solid State Physics. Saunders College Publishing, Orlando (1976).
Ball, J. and James, R., Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13-52. CrossRef
Ball, J. and James, R., Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. Lond. A 338 (1992) 389-450. CrossRef
Bernadou, M. and Hassan, K., Basis functions for general Hsieh-Clough-Tocher triangles, complete or reduced. Internat. J. Numer. Methods Engrg. 17 (1981) 784-789. CrossRef
Bhattacharya, K. and James, R.D., A theory of thin films of martensitic materials with applications to microactuators. J. Mech. Phys. Solids 47 (1999) 531-576. CrossRef
Bhattacharya, K., Li, B. and Luskin, M., The simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation. Arch. Rat. Mech. Anal. 149 (1999) 123-154. CrossRef
S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer-Verlag, New York (1994).
P. Belík, T. Brule and M. Luskin, Numerical modelling of a temperature-operated martensitic microvalve. http://www.math.umn.edu/ luskin/research/valve/.
Belík, P. and Luskin, M., Stability of microstructure for tetragonal to monoclinic martensitic transformations. ESAIM: M2AN 34 (2000) 663-685. CrossRef
Carstensen, C. and Plechác, P., Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. 66 (1997) 997-1026. CrossRef
Carstensen, C. and Plechác, P., Adaptive algorithms for scalar non-convex variational problems. Appl. Numer. Math. 26 (1998) 203-216. CrossRef
Chipot, M., Collins, C. and Kinderlehrer, D., Numerical analysis of oscillations in multiple well problems. Numer. Math. 70 (1995) 259-282. CrossRef
Chipot, M. and Kinderlehrer, D., Equilibrium configurations of crystals. Arch. Rat. Mech. Anal. 103 (1988) 237-277. CrossRef
M. Chipot and S. Müller, Sharp energy estimates for finite element approximations of nonconvex problems. Preprint (1997).
P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978).
R.W. Clough and J.L. Tocher, Finite element stiffness matrices for analysis of plates in bending. In Proceedings of the conference on matrix methods in structural mechanics. Wright Patterson A.F.B., Ohio (1965) 515-545.
C. Collins, Computation of twinning. In Microstructure and phase transitions. J. Ericksen, R. James, D. Kinderlehrer and M. Luskin Eds. IMA Vol. Math. Applic. 54, Springer-Verlag, New York (1993) 39-50.
C. Collins and M. Luskin, The computation of the austenitic-martensitic phase transition. In Partial differential equations and continuum models of phase transitions. M. Rascle, D. Serre and M. Slemrod Eds. Lect. Notes Phys. 344, Springer-Verlag, Berlin (1989) 34-50.
C. Collins, M. Luskin and J. Riordan, Computational results for a two-dimensional model of crystalline microstructure. In Microstructure and phase transitions. J. Ericksen, R. James, D. Kinderlehrer and M. Luskin Eds. IMA Vol. Math. Applic. 54, Springer-Verlag, New York (1993) 51-56.
Dolzmann, G., Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal. 36 (1999) 1621-1635. CrossRef
Dong, J.W., Chen, L.C., Palmstrøm, C.J., James, R.D. and McKernann, S., Molecular beam epitaxy growth of ferromagnetic single crystal (001) Ni2MnGa on (001) GaAs. Appl. Phys. Lett. 75 (1999) 1443-45. CrossRef
Dunavant, D.A., k High degree efficient symmetrical Gaussian quadrature rules for the triangle. Internat. J. Numer. Methods Engrg. 21 (1985) 1129-1148. CrossRef
G.B. Folland, Real analysis. Modern techniques and their applications. John Wiley & Sons, Inc., New York (1984).
M. Giaquinta. Calculus of variations. Springer-Verlag, Berlin (1996).
D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin (1998).
R. Glowinski, Numerical methods for nonlinear variational problems. Springer-Verlag, New York (1984).
Gremaud, P.-A., Numerical analysis of a nonconvex variational problem related to solid-solid phase transitions. SIAM J. Numer. Anal. 31 (1994) 111-127. CrossRef
M. Gurtin, Topics in finite elasticity. SIAM, Philadelphia (1981).
R.D. James and R. Rizzoni, Pressurized shape memory thin films. J. Elasticity 59, special issue in honor of Roger Fosdick, D. Carlson Ed. (2000) 399-436.
Krulevitch, P., Lee, A.P., Ramsey, P.B., Trevino, J.C., Hamilton, J. and Northrup, M.A., Thin film shape memory alloy microactuators. Journal of Microelectromechanical Systems 5 (1996) 270. CrossRef
Kruzík, M., Numerical approach to double well problems. SIAM J. Numer. Anal. 35 (1998) 1833-1849. CrossRef
P. Lascaux and P. Lesaint, Some nonconforming finite elements for the plate bending problem. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. R-1 (1975) 9-53.
Li, B. and Luskin, M., Finite element analysis of microstructure for the cubic to tetragonal transformation. SIAM J. Numer. Anal. 35 (1998) 376-392. CrossRef
Li, B. and Luskin, M., Nonconforming finite element approximation of crystalline microstructure. Math. Comp. 67 (1998) 917-946. CrossRef
Li, B. and Luskin, M., Approximation of a martensitic laminate with varying volume fractions. ESAIM: M2AN 33 (1999) 67-87. CrossRef
Simultaneous, Z. Li numerical approximation of microstructures and relaxed minimizers. Numer. Math. 78 (1997) 21-38.
D.G. Luenberger, Introduction to linear and nonlinear programming. Addison-Wesley, Reading, Mass. (1973).
Luskin, M., Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math. 75 (1996) 205-221. CrossRef
Luskin, M., On the computation of crystalline microstructure. Acta Numer. 5 (1996) 191-257. CrossRef
Luskin, M. and Analysis, L. Ma of the finite element approximation of microstructure in micromagnetics. SIAM J. Numer. Anal. 29 (1992) 320-331. CrossRef
Morley, L.S.D., The triangular equilibrium element in the solution of plate bending problems. Aero. Quart. 19 (1968) 149-169.
Pedregal, P., On the numerical analysis of non-convex variational problems. Numer. Math. 74 (1996) 325-336. CrossRef
E. Polak, Computational methods in optimization. Academic Press, New York (1971).
Roubícek, T., Numerical approximation of relaxed variational problems. J. Convex Anal. 3 (1996) 329-347.
H.L. Royden, Real analysis. 3rd edn, Macmillan Publishing Company, New York (1988).
W. Rudin, Functional analysis. McGraw-Hill, New York (1973).
Shi, Z., Error estimates of Morley element. Chinese J. Num. Math. Appl. 12 (1990) 102-108.