The eigenvalue problem for a class of long, thin elastic structures with periodic geometry

Author:
Robert E. Miller

Journal:
Quart. Appl. Math. **52** (1994), 261-282

MSC:
Primary 73B27; Secondary 35B27, 73K05, 73K12, 73K20

DOI:
https://doi.org/10.1090/qam/1276237

MathSciNet review:
MR1276237

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Elastic structures consisting of many thin elements arranged periodically (such as grids and trusses) are common in applications. Using standard numerical techniques such as splines in attempting to analyze these structures leads to serious difficulties due to the complicated geometry. Instead, one can use methods of asymptotic analysis to derive a ``simple'' problem whose solution approximates that of the original problem. In this paper we begin with a linearized elastic system on a three-dimensional domain with two of the dimensions small relative to the third and derive a one-dimensional eigenvalue problem by letting a small parameter tend to zero. The resulting equation has coefficients which vary periodically with the spatial variable, so we let the period tend to zero to obtain the ``homogenized'' equation which has constant coefficients whose values can be easily calculated once the geometry of the structure is specified. We illustrate with several examples.

**[1]**A. Bensoussan, J. L. Lions, and G. Papanicolau,*Asymptotic Analysis for Periodic Structures*, North-Holland, Amsterdam, 1978**[2]**P. G. Ciarlet,*Mathematical Elasticity*, Vol. 1:*Three-Dimensional Elasticity*, North-Holland, Amsterdam, 1988**[3]**P. G. Ciarlet,*Plates and Junctions in Elastic Multi-Structures*:*An Asymptotic Analysis*, Collection Recherches en Mathématiques Appliquées, vol. 14, Masson, Paris, 1990**[4]**P. G. Ciarlet and S. Kesavan,*Two-dimensional approximations of three-dimensional eigenvalue problems in plate theory*, Comput. Methods Appl. Mech. Engrg.**26**, 145-172 (1981)**[5]**A. Cimetière et al.,*Asymptotic theory and analysis for displacements and stress distribution in nonlinear elastic straight slender rods*, J. Elasticity**19**, 111-161 (1988)**[6]**D. Cioranescu and J. Saint Jean Paulin,*Homogenization in open sets with holes*, J. Math. Anal. Appl.**71**, 590-607 (1979)**[7]**D. Cioranescu and J. Saint Jean Paulin,*Reinforced and honey-comb structures*, J. Math. Pures Appl.**65**, 403-422 (1986)**[8]**D. Cioranescu and J. Saint Jean Paulin,*Elasticity problems for towers*, preprint**[9]**D. Cioranescu and J. Saint Jean Paulin,*Towers and cranes in linearized elasticity*:*An asymptotic study*, preprint**[10]**S. Kesavan,*Homogenization of elliptic eigenvalue problems*. 1, Appl. Math. Optim.**5**, 153-167 (1979)**[11]**J. L. Lions,*Some Methods in the Mathematical Analysis of Systems and Their Control*, Gordon and Breach, Science Press, Bejing and New York, 1981**[12]**J. Sanchez-Hubert and E. Sanchez-Palencia,*Vibration and Coupling of Continuous Systems*:*Asymptotic Methods*, Springer-Verlag, Berlin, 1989**[13]**E. Sanchez-Palencia,*Non-Homogeneous Media and Vibration Theory*, Lecture Notes in Phys., vol. 127, Springer-Verlag, Berlin, 1980**[14]**E. Sanchez-Palencia and A. Zaoui, eds.,*Homogenization Techniques for Composite Media*, Lecture Notes in Phys., vol. 272, Springer-Verlag, Berlin, 1987**[15]**H. F. Weinberger,*Variational Methods for Eigenvalue Approximation*, SIAM, Philadelphia, PA, 1974**[16]**J. Wloka,*Partial Differential Equations*, Cambridge Univ. Press, Cambridge, 1987

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
73B27,
35B27,
73K05,
73K12,
73K20

Retrieve articles in all journals with MSC: 73B27, 35B27, 73K05, 73K12, 73K20

Additional Information

DOI:
https://doi.org/10.1090/qam/1276237

Article copyright:
© Copyright 1994
American Mathematical Society