On the numerical modeling of deformations of pressurized martensitic thin films

Bělík, Pavel; Brule, Timothy; Luskin, Mitchell

ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 3, pp. 525-548.

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.

[1] R. Adams, Sobolev spaces. Academic Press, New York (1975). | MR | Zbl

[2] J.H. Argyris, I. Fried and D.W. Scharpf, The TUBA family of plate elements for the matrix displacement method. Aero. J. Roy. Aero. Soc. 72 (1968) 701-709.

[3] N.W. Ashcroft and N.D. Mermin, Solid State Physics. Saunders College Publishing, Orlando (1976).

[4] J. Ball and R. James, Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13-52. | Zbl

[5] J. Ball and R. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. Lond. A 338 (1992) 389-450. | Zbl

[6] M. Bernadou and K. Hassan, Basis functions for general Hsieh-Clough-Tocher triangles, complete or reduced. Internat. J. Numer. Methods Engrg. 17 (1981) 784-789. | Zbl

[7] K. Bhattacharya and R.D. James, A theory of thin films of martensitic materials with applications to microactuators. J. Mech. Phys. Solids 47 (1999) 531-576. | Zbl

[8] K. Bhattacharya, B. Li and M. Luskin, The simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation. Arch. Rat. Mech. Anal. 149 (1999) 123-154. | Zbl

[9] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer-Verlag, New York (1994). | MR | Zbl

[10] P. Bělík, T. Brule and M. Luskin, Numerical modelling of a temperature-operated martensitic microvalve. http://www.math.umn.edu/~luskin/research/valve/.

[11] P. Bělík and M. Luskin, Stability of microstructure for tetragonal to monoclinic martensitic transformations. ESAIM: M2AN 34 (2000) 663-685. | Numdam | Zbl

[12] C. Carstensen and P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. 66 (1997) 997-1026. | Zbl

[13] C. Carstensen and P. Plecháč, Adaptive algorithms for scalar non-convex variational problems. Appl. Numer. Math. 26 (1998) 203-216. | Zbl

[14] M. Chipot, C. Collins and D. Kinderlehrer, Numerical analysis of oscillations in multiple well problems. Numer. Math. 70 (1995) 259-282. | Zbl

[15] M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rat. Mech. Anal. 103 (1988) 237-277. | Zbl

[16] M. Chipot and S. Müller, Sharp energy estimates for finite element approximations of nonconvex problems. Preprint (1997).

[17] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | MR | Zbl

[18] 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.

[19] 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. | Zbl

[20] 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. | Zbl

[21] 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. | Zbl

[22] G. Dolzmann, Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal. 36 (1999) 1621-1635. | Zbl

[23] J.W. Dong, L.C. Chen, C.J. Palmstrøm, R.D. James and S. Mckernann, Molecular beam epitaxy growth of ferromagnetic single crystal (001) Ni${}_2$MnGa on (001) GaAs. Appl. Phys. Lett. 75 (1999) 1443-45.

[24] D.A. Dunavant, k High degree efficient symmetrical Gaussian quadrature rules for the triangle. Internat. J. Numer. Methods Engrg. 21 (1985) 1129-1148. | Zbl

[25] G.B. Folland, Real analysis. Modern techniques and their applications. John Wiley & Sons, Inc., New York (1984). | MR | Zbl

[26] M. Giaquinta. Calculus of variations. Springer-Verlag, Berlin (1996). | Zbl

[27] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin (1998). | Zbl

[28] R. Glowinski, Numerical methods for nonlinear variational problems. Springer-Verlag, New York (1984). | MR | Zbl

[29] P.-A. Gremaud, Numerical analysis of a nonconvex variational problem related to solid-solid phase transitions. SIAM J. Numer. Anal. 31 (1994) 111-127. | Zbl

[30] M. Gurtin, Topics in finite elasticity. SIAM, Philadelphia (1981). | MR

[31] 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. | Zbl

[32] P. Krulevitch, A.P. Lee, P.B. Ramsey, J.C. Trevino, J. Hamilton and M.A. Northrup, Thin film shape memory alloy microactuators. Journal of Microelectromechanical Systems 5 (1996) 270.

[33] M. Kružík, Numerical approach to double well problems. SIAM J. Numer. Anal. 35 (1998) 1833-1849. | Zbl

[34] 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. | Numdam | Zbl

[35] B. Li and M. Luskin, Finite element analysis of microstructure for the cubic to tetragonal transformation. SIAM J. Numer. Anal. 35 (1998) 376-392. | Zbl

[36] B. Li and M. Luskin, Nonconforming finite element approximation of crystalline microstructure. Math. Comp. 67 (1998) 917-946. | Zbl

[37] B. Li and M. Luskin, Approximation of a martensitic laminate with varying volume fractions. ESAIM: M2AN 33 (1999) 67-87. | Numdam | Zbl

[38] Z. Li, Simultaneous numerical approximation of microstructures and relaxed minimizers. Numer. Math. 78 (1997) 21-38. | Zbl

[39] D.G. Luenberger, Introduction to linear and nonlinear programming. Addison-Wesley, Reading, Mass. (1973). | Zbl

[40] M. Luskin, Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math. 75 (1996) 205-221. | Zbl

[41] M. Luskin, On the computation of crystalline microstructure. Acta Numer. 5 (1996) 191-257. | Zbl

[42] M. Luskin and L. Ma, Analysis of the finite element approximation of microstructure in micromagnetics. SIAM J. Numer. Anal. 29 (1992) 320-331. | Zbl

[43] L.S.D. Morley, The triangular equilibrium element in the solution of plate bending problems. Aero. Quart. 19 (1968) 149-169.

[44] P. Pedregal, On the numerical analysis of non-convex variational problems. Numer. Math. 74 (1996) 325-336. | Zbl

[45] E. Polak, Computational methods in optimization. Academic Press, New York (1971). | MR

[46] T. Roubíček, Numerical approximation of relaxed variational problems. J. Convex Anal. 3 (1996) 329-347. | Zbl

[47] H.L. Royden, Real analysis. 3rd edn, Macmillan Publishing Company, New York (1988). | MR | Zbl

[48] W. Rudin, Functional analysis. McGraw-Hill, New York (1973). | MR | Zbl

[49] Z. Shi, Error estimates of Morley element. Chinese J. Num. Math. Appl. 12 (1990) 102-108.