메뉴 건너뛰기




Volumn 34, Issue 3, 2000, Pages 663-685

Stability of microstructure for tetragonal to monoclinic martensitic transformations

Author keywords

Error estimate; Finite element; Martensitic transformation; Microstructure; Monoclinic; Nonconvex variational problem; Simple laminate; Tetragonal; Volume fraction; Young measure

Indexed keywords


EID: 0034396221     PISSN: 0764583X     EISSN: None     Source Type: Journal    
DOI: 10.1051/m2an:2000161     Document Type: Article
Times cited : (9)

References (38)
  • 1
    • 0003796630 scopus 로고
    • Academic Press, New York
    • R. Adams, Sobolev Spaces. Academic Press, New York (1975).
    • (1975) Sobolev Spaces
    • Adams, R.1
  • 2
    • 33750297145 scopus 로고
    • Fine phase mixtures as minimizers of energy
    • J. Ball and R. James, Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13-52.
    • (1987) Arch. Rat. Mech. Anal. , vol.100 , pp. 13-52
    • Ball, J.1    James, R.2
  • 3
    • 0002854369 scopus 로고
    • Proposed experimental tests of a theory of fine microstructure and the two-well problem
    • 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.
    • (1992) Phil. Trans. R. Soc. Lond. A , vol.338 , pp. 389-450
    • Ball, J.1    James, R.2
  • 4
    • 0345406326 scopus 로고
    • Self accomodation in martensite
    • K. Bhattacharya, Self accomodation in martensite. Arch. Rat. Mech. Anal. 120 (1992) 201-244.
    • (1992) Arch. Rat. Mech. Anal. , vol.120 , pp. 201-244
    • Bhattacharya, K.1
  • 6
    • 0033437538 scopus 로고    scopus 로고
    • The simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation
    • 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 (2000) 123-154.
    • (2000) Arch. Rat. Mech. Anal. , vol.149 , pp. 123-154
    • Bhattacharya, K.1    Li, B.2    Luskin, M.3
  • 7
    • 0032184250 scopus 로고    scopus 로고
    • Approximation of infima in the calculus of variations
    • B. Brighi and M. Chipot, Approximation of infima in the calculus of variations. J. Comput. Appl. Math. 98 (1998) 273-287.
    • (1998) J. Comput. Appl. Math. , vol.98 , pp. 273-287
    • Brighi, B.1    Chipot, M.2
  • 8
    • 0031533725 scopus 로고    scopus 로고
    • Numerical solution of the scalar double-well problem allowing microstructure
    • C. Carstensen and P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp., 66 (1997) 997-1026.
    • (1997) Math. Comp. , vol.66 , pp. 997-1026
    • Carstensen, C.1    Plecháč, P.2
  • 9
    • 0031644439 scopus 로고    scopus 로고
    • Adaptive algorithms for scalar non-convex variational problems
    • C. Carstensen and P. Plecháč, Adaptive algorithms for scalar non-convex variational problems. Appl. Numer. Math. 26 (1998) 203-216.
    • (1998) Appl. Numer. Math. , vol.26 , pp. 203-216
    • Carstensen, C.1    Plecháč, P.2
  • 10
    • 0001006601 scopus 로고
    • Numerical analysis of oscillations in nonconvex problems
    • M. Chipot, Numerical analysis of oscillations in nonconvex problems. Numer. Math. 59 (1991) 747-767.
    • (1991) Numer. Math. , vol.59 , pp. 747-767
    • Chipot, M.1
  • 11
    • 0026907665 scopus 로고
    • Numerical approximations in variational problems with potential wells
    • M. Chipot and C. Collins, Numerical approximations in variational problems with potential wells. SIAM J. Numer. Anal. 29 (1992) 1002-1019.
    • (1992) SIAM J. Numer. Anal. , vol.29 , pp. 1002-1019
    • Chipot, M.1    Collins, C.2
  • 12
    • 21844504886 scopus 로고
    • Numerical analysis of oscillations in multiple well problems
    • M. Chipot, C. Collins, and D. Kinderlehrer, Numerical analysis of oscillations in multiple well problems. Numer. Math. 70 (1995) 259-282.
    • (1995) Numer. Math. , vol.70 , pp. 259-282
    • Chipot, M.1    Collins, C.2    Kinderlehrer, D.3
  • 13
  • 15
    • 0026136907 scopus 로고
    • Numerical approximation of the solution of a variational problem with a double well potential
    • C. Collins, D. Kinderlehrer, and M. Luskin, Numerical approximation of the solution of a variational problem with a double well potential. SIAM J. Numer. Anal. 28 (1991) 321-332.
    • (1991) SIAM J. Numer. Anal. , vol.28 , pp. 321-332
    • Collins, C.1    Kinderlehrer, D.2    Luskin, M.3
  • 16
    • 84966255684 scopus 로고
    • Optimal order estimates for the finite element approximation of the solution of a nonconvex variational problem
    • C. Collins and M. Luskin, Optimal order estimates for the finite element approximation of the solution of a nonconvex variational problem. Math. Comp. 57 (1991) 621-637.
    • (1991) Math. Comp. , vol.57 , pp. 621-637
    • Collins, C.1    Luskin, M.2
  • 18
    • 0001459333 scopus 로고    scopus 로고
    • Numerical computation of rank-one convex envelopes
    • G. Dolzmann, Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal. 36 (1999) 1621-1635.
    • (1999) SIAM J. Numer. Anal. , vol.36 , pp. 1621-1635
    • Dolzmann, G.1
  • 19
    • 0025416425 scopus 로고
    • On the convergence of finite element approximations of a relaxed variational problem
    • D. French, On the convergence of finite element approximations of a relaxed variational problem. SIAM J. Numer. Anal. 28 (1991) 419-436.
    • (1991) SIAM J. Numer. Anal. , vol.28 , pp. 419-436
    • French, D.1
  • 20
    • 0031258822 scopus 로고    scopus 로고
    • 4 by energy minimization
    • 4 by energy minimization. Acta Mater. 45 (1997) 4271-4281.
    • (1997) Acta Mater. , vol.45 , pp. 4271-4281
    • Jian, L.1    James, R.2
  • 21
    • 0001139965 scopus 로고
    • Characterizations of gradient Young measures
    • D. Kinderlehrer and P. Pedregal, Characterizations of gradient Young measures. Arch. Rat. Mech. Anal. 115 (1991) 329-365.
    • (1991) Arch. Rat. Mech. Anal. , vol.115 , pp. 329-365
    • Kinderlehrer, D.1    Pedregal, P.2
  • 22
    • 0000635099 scopus 로고    scopus 로고
    • Numerical approach to double well problems
    • M. Kružík, Numerical approach to double well problems. SIAM J. Numer. Anal. 35 (1998) 1833-1849.
    • (1998) SIAM J. Numer. Anal. , vol.35 , pp. 1833-1849
    • Kružík, M.1
  • 23
    • 0040170655 scopus 로고    scopus 로고
    • Finite element analysis of microstructure for the cubic to tetragonal transformation
    • B. Li and M. Luskin, Finite element analysis of microstructure for the cubic to tetragonal transformation. SIAM J. Numer. Anal. 35 (1998) 376-392.
    • (1998) SIAM J. Numer. Anal. , vol.35 , pp. 376-392
    • Li, B.1    Luskin, M.2
  • 24
    • 0032364336 scopus 로고    scopus 로고
    • Nonconforming finite element approximation of crystalline microstructure
    • B. Li and M. Luskin, Nonconforming finite element approximation of crystalline microstructure. Math. Comp. 67(223) (1998) 917-946.
    • (1998) Math. Comp. , vol.67 , Issue.223 , pp. 917-946
    • Li, B.1    Luskin, M.2
  • 25
    • 0033456348 scopus 로고    scopus 로고
    • Approximation of a martensitic laminate with varying volume fractions
    • B. Li and M. Luskin, Approximation of a martensitic laminate with varying volume fractions. Math. Model. Numer. Anal. 33 (1999) 67-87.
    • (1999) Math. Model. Numer. Anal. , vol.33 , pp. 67-87
    • Li, B.1    Luskin, M.2
  • 26
    • 0031322111 scopus 로고    scopus 로고
    • Simultaneous numerical approximation of microstructures and relaxed minimizers
    • Z. Li, Simultaneous numerical approximation of microstructures and relaxed minimizers. Numer. Math. 78 (1997) 21-38.
    • (1997) Numer. Math. , vol.78 , pp. 21-38
    • Li, Z.1
  • 27
    • 0030353296 scopus 로고    scopus 로고
    • Approximation of a laminated microstructure for a rotationally invariant, double well energy density
    • M. Luskin, Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math. 75 (1996) 205-221.
    • (1996) Numer. Math. , vol.75 , pp. 205-221
    • Luskin, M.1
  • 28
    • 0347284844 scopus 로고    scopus 로고
    • On the computation of crystalline microstructure
    • M. Luskin, On the computation of crystalline microstructure. Acta Numer. (1996) 191-257.
    • (1996) Acta Numer. , pp. 191-257
    • Luskin, M.1
  • 29
    • 0026843542 scopus 로고    scopus 로고
    • Analysis of the finite element approximation of microstructure in micromagnetics
    • M. Luskin and L. Ma, Analysis of the finite element approximation of microstructure in micromagnetics. SIAM J. Numer. Anal. 29 320-331.
    • SIAM J. Numer. Anal. , vol.29 , pp. 320-331
    • Luskin, M.1    Ma, L.2
  • 30
    • 84968511032 scopus 로고
    • Strong convergence of numerical solutions to degenerate variational problems
    • R. Nicolaides and N. Walkington, Strong convergence of numerical solutions to degenerate variational problems. Math. Comp. 64 (1995) 117-127.
    • (1995) Math. Comp. , vol.64 , pp. 117-127
    • Nicolaides, R.1    Walkington, N.2
  • 31
    • 0000572269 scopus 로고
    • Numerical approximation of parametrized measures
    • P. Pedregal, Numerical approximation of parametrized measures. Num. Funct. Anal. Opt. 16 (1995) 1049-1066.
    • (1995) Num. Funct. Anal. Opt. , vol.16 , pp. 1049-1066
    • Pedregal, P.1
  • 32
    • 0030508310 scopus 로고    scopus 로고
    • On the numerical analysis of non-convex variational problems
    • P. Pedregal, On the numerical analysis of non-convex variational problems. Numer. Math. 74 (1996) 325-336.
    • (1996) Numer. Math. , vol.74 , pp. 325-336
    • Pedregal, P.1
  • 33
    • 21744444554 scopus 로고    scopus 로고
    • Numerical approximation of relaxed variational problems
    • T. Roubíček, Numerical approximation of relaxed variational problems. J. Convex Anal. 3 (1996) 329-347.
    • (1996) J. Convex Anal. , vol.3 , pp. 329-347
    • Roubíček, T.1
  • 34
    • 33748684833 scopus 로고
    • Crystallography of the tetragonal → monoclinic transformation in zirconia
    • N. Simha, Crystallography of the tetragonal → monoclinic transformation in zirconia. J. Phys. IV Colloq. France 5 (1995) C81121-C81126.
    • (1995) J. Phys. IV Colloq. France , vol.5
    • Simha, N.1
  • 35
    • 0031062025 scopus 로고    scopus 로고
    • Twin and habit plane microstructures due to the tetragonal to monoclinic transformation of zirconia
    • N. Simha, Twin and habit plane microstructures due to the tetragonal to monoclinic transformation of zirconia. J. Mech. Phys. Solids 45 (1997) 261-292.
    • (1997) J. Mech. Phys. Solids , vol.45 , pp. 261-292
    • Simha, N.1
  • 36
    • 0038831167 scopus 로고
    • Lower-semicontinuity of variational integrals and compensated compactness
    • Zürich Birkhäuser
    • V. Šverák, Lower-semicontinuity of variational integrals and compensated compactness, in Proceedings ICM 94, Zürich (1995). Birkhäuser.
    • (1995) Proceedings ICM 94
    • Šverák, V.1
  • 37
    • 0000440966 scopus 로고
    • Compensated compactness and applications to partial differential equations
    • Nonlinear analysis and mechanics, R. Knops, Ed.
    • L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics, R. Knops, Ed., Pitman Research Notes in Mathematics, London 39 (1978) 136-212.
    • (1978) Pitman Research Notes in Mathematics, London , vol.39 , pp. 136-212
    • Tartar, L.1
  • 38
    • 0010485120 scopus 로고
    • Twinning in minerals and metals: Remarks on the comparison of a thermoelasticity theory with some available experimental results
    • G. Zanzotto, Twinning in minerals and metals: remarks on the comparison of a thermoelasticity theory with some available experimental results. Atti Acc. Lincei Rend. Fis. 82 (1988) 725-756.
    • (1988) Atti Acc. Lincei Rend. Fis. , vol.82 , pp. 725-756
    • Zanzotto, G.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.