Summary
The state of stress and deformation of a planar elastic-homogeneous transversely isotropic thick layer in the case of plane and axisymmetric strain respectively is determined in a systematic and uniform manner using integral transforms and transfer matrices. Next, a laminate with an arbitrary number of different layers is considered without any simplifying assumptions. Then we analyze a periodic structure consisting of many thin and identical layer groups by means of a suitable homogenization, where a layer group contains two or more different transversely isotropic homogeneous basic layers. As an example exact closed form solutions for a periodically layered half space are evaluated. The well known result, that a medium which is composed of alternating thin layers of two different elastic-isotropic substances is elastostatically equivalent to a homogeneous transversely isotropic medium is extended to the above mentioned more general case. Further, the in-plane normal stresses which are discontinuous for a finite layering are evaluated in addition to the smeared ones which are continuous and correct only in the limit of a vanishing thickness of the individual layers. The explicit knowledge of the resultant elastic constants (called effective ones) turns out to be dispensable; rather, the effective material parameters (defined in this paper) which are the weighted sums of the material parameters of the basic layers are proved to be relevant. Nevertheless, for the purpose of comparison with some published results the effective elastic constants, especially for a layer group consisting of two different elastic-isotropic substances, are evaluated additionally.
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Dedicated to Prof. Gallus Rehm on the occasion of his 75th birthday
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Bufler, H. Planar elastic laminates and their homogenization. Acta Mechanica 141, 21–36 (2000). https://doi.org/10.1007/BF01176805
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DOI: https://doi.org/10.1007/BF01176805