Homogenization limits of the equations of elasticity in thin domains.

*(English)*Zbl 0614.73012In this paper the authors studied pure bending of a flat, linearly elastic three-dimensional plate with rapidly varying composition. In addition to the constitutive elastic law appropriate uniform coercivity and boundedness conditions are placed, but no special structure in composition such as periodicity or quasiperiodicity is required. Under these assumptions it is shown that all limiting vertical displacements, coming from the equations of three-dimensional elasticity with plate thickness approaching zero, must necessarily satisfy a fourth order, two- dimensional plate equation on the plate midplane. It is worthy of mentioning here that the plates with rapidly varying composition which are considered in this paper are of interest in structural optimization. In certain design contexts they are known to be stronger than plates with only slow variation in composition. A more detailed discussion of the relation between an optimal design problem and plates with rapidly varying composition is given by other authors.

Reviewer: W.A.Bassali

##### MSC:

74E05 | Inhomogeneity in solid mechanics |

74K20 | Plates |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

74P99 | Optimization problems in solid mechanics |

74E30 | Composite and mixture properties |