Article
Full entry |
PDF
(2.4 MB)
Feedback
PDF
(2.4 MB)
Feedback
Keywords:
homogenization; approximation of material with periodic structure by homogeneous one; terms of displacements; terms of stresses; results compared; multiple-scale method; properties of homogenized coefficients; correctors; convergence of displacement vector; stress tensor; local energy; simplified local energy method
homogenization; approximation of material with periodic structure by homogeneous one; terms of displacements; terms of stresses; results compared; multiple-scale method; properties of homogenized coefficients; correctors; convergence of displacement vector; stress tensor; local energy; simplified local energy method
Summary:
The homogenization problem (i.e. the approximation of the material with periodic structure by a homogeneous one) for linear elasticity equation is studied. Both formulations in terms of displacements and in terms of stresses are considered and the results compared. The homogenized equations are derived by the multiple-scale method. Various formulae, properties of the homogenized coefficients and correctors are introduced. The convergence of displacment vector, stress tensor and local energy is proved by a simplified local energy method.
References:
[1] A. Ambrosetti C. Sbordone: $\Gamma$-convergenza e G-convergenza per problemi non lineari di tipo ellittici. Bol. Un. Mat. Ital. A(5), 13 (1976), 352-362. MR 0487703
[2] I. Babuška: Solution of interface problems by homogenization I, II, III. SIAM J. Math. Anal., 7(1976), 603-634 (I), 635-645 (II), 8(1977), 923-937 (III). DOI 10.1137/0507048 | MR 0509273
[3] I. Babuška: Homogenization and its application. Mathematical and computational problems. Numerical solution of partial differential equations, III. (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975), 89-116, Academic Press, New York, 1976. MR 0502025
[4] N. S. Bahvalov: The averaging of partial differential equations with rapidly oscillating coefficients. (Russian) Problems in mathematical physics and numerical mathematics (Russian), 34-51, 323, "Nauka", Moscow, 1977. MR 0521167
[5] A. Bensoussan J. L. Lions G. Papanicolaou: Asymptotic analysis for periodic structures. North Holland 1978. MR 0503330
[6] V. L. Berdičevskij: On averaging of periodic structures. (Russian), Prikl. Mat. Meh., 41 (1977), 6, 993-1006. MR 0529542
[7] M. Biroli: G-convergence for elliptic equations, variational inequalities and quasivariational inequalities. Rend. Sem. Mat. Fis. Milano, 47 (1977), 269 - 328. DOI 10.1007/BF02925757 | MR 0526888
[8] J. F. Bourgat: Numerical experiments of the homogenization method for operators with periodic coefficients. IRIA-LABORIA Report, no. 277 (1978); Computing methods in applied sciences and engineering (Proc. Third Internat. Sympos., Versailles, 1977), I, 330-356, Lecture Notes in Math., 704, Springer, Berlin, 1979. MR 0540121
[9] J. F. Bourgat A. Dervieux: Méthode d'homogénéisation de opérateurs à coefficients périodiques: Etude des correcteurs provenant du développement asymptotique. IRIA-LABORIA Report, n. 278 (1978).
[10] E. De Giorgi: Convergence problems for functional and operators. Proceedings of the International Meeting on Recent Methods in Non-linear Analysis (Rome, 1978), 131 - 188, Pitagora, Bologna, 1979. MR 0533166
[11] P. Marcellini: Periodic solutions and homogenization of nonlinear variational problems. Ann. Mat. Appl. (4). 117 (1978), 139-152. MR 0515958
[12] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague 1967. MR 0227584
[13] J. Nečas I. Hlaváček: Mathematical theory of elastic and elastico-plastic bodies: An introduction. Elsevier, Amsterdam 1981. MR 0600655
[14] Ha Tien Ngoan: On convergence of solutions of boundary value problems for sequence of elliptic systems. (Russian), Vestnik Moskov. Univ. Ser. I Mat. Meh., 5 (1977), 83 - 92.
[15] E. Sanchez Palencia: Comportements local et macroscopique d'un type de milieux physiques hétérogènes. Internát. J. Engrg. ScL, 12 (1974), 331 - 351. DOI 10.1016/0020-7225(74)90062-7 | MR 0441059 | Zbl 0275.76032
[16] S. Spagnolo: Convergence in energy for elliptic operators. Numerical solution of partial differential equations, III. (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975), 469-498. Academic Press, New York, 1976. MR 0477444
[17] P. M. Suquet: Une méthode duále en homogénéisation. Application aux milieux élastiques périodiques. C. R. Acad. Sci. Paris Sér. A, 291 (1980), 181 - 184. MR 0605012 | Zbl 0491.73024
[18] V. V. Žikov S. M. Kozlov O. A. Olejnik, Ha Tien Ngoan: Homogenization and G-convergence of differential operators. (Russian), Uspehi Mat. Nauk, 34 (1979), 5 (209), 65-133. MR 0562800
[19] P. M. Suquet: Une méthode duále en homogénéisation: Application aux milieux élastiques. Submitted to J. Mécanique. Zbl 0516.73016
Search
Partner of
