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CivilComp Proceedings
ISSN 17593433 CCP: 75
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and Z. Bittnar
Paper 90
A Unified Formulation for Multilayered Plate Elements based on PVD and RMVT E. Carrera
Aerospace Department, Politecnico di Torino, Italy E. Carrera, "A Unified Formulation for Multilayered Plate Elements based on PVD and RMVT", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Sixth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 90, 2002. doi:10.4203/ccp.75.90
Keywords: multilayered plates, refined theories, mixed methods, finite element, zigzag, interlaminar continuity, transverse normal stress.
Summary
Multilayered made structures are increasingly used in aerospace,
ships, automotive vehicles, advanced optical mirrors
and semiconductor technologies.
In most of the applications, these structures mostly appear as flat (plates)
or curved panels (shells).
In this work, attention has been restricted to flat geometries,
although most of the presented derivations and techniques could be
extended to shell cases.
The analysis of multilayered, anisotropic structures is difficult when compared
to one layered ones made of traditional isotropic materials.
A number of complicating effects arise when their mechanical behavior as well
as failure mechanisms have to be correctly understood.
As far as two dimensional modeling is concerned, the subject to
which this paper is devoted, layered structures also require special attention.
This is due to the intrinsic discontinuity of the thermomechanical
properties at each layerinterface to which high shear and normal
transverse deformabilty is associated.
An accurate description of the stress and strain fields of these structures
requires theories that are able to describe the socalled ZigZag (ZZ) form of displacement
fields in the thickness direction as well as Interlaminar Continuous (IC)
transverse shear and normal stresses.
These two facts have been summarized in [1] as Requirements,
that is displacements and transverse stress components must be function in the
thickness direction .
Transverse and inplane anisotropy of multilayered structures along with
the fulfillment of the Requirements
make it difficult to solve practical problems related to
layered structures.
The use of both refined twodimensional theories
and computational methods become mandatory
to solve practical problems related
to multilayered structures.
Among the several available computational methods,
the Finite Element Method (FEM) has played and continues to play a significant role.
Most of the commercial codes that are used in small and large
companies as well as in Research Centers are, in fact, finite element oriented.
A large number of refined theories and computational strategies have been proposed and implemented over the last four decades. Many review works are available on these subject. Among these, recommended reviews are those quoted in the articles [2,3]. The subjected of the present work is that of multilayered finite elements that are able to furnish an accurate description of strain/stress fields in multilayer flat structure analysis. Recent author's findings [1], [2],[5] are herein employed to formulate in unified manner a large variety of finite elements. Reissner's Variational Mixed Theorem, RMVT, and Principle of Virtual Displacements, PVD, have been used to derive advanced(*) and classical multilayered finite elements, respectively. The number of both the order of expansion in and the number of nodes of the elements, are taken as free parameters of the considered RMVT and PVD formulations. Layer stiffness are accumulated at plate as well as layer level by permitting than Equivalent Single Layer, ESLM, as well as LayerWise, LW, variable description. In order to lower the number of the equations related to the several presented finite elements as much as possible, an indicial notation has been used. As a fundamental property such an indicial notation has led to the writing of all the finite element matrices in terms of a few arrays, which are called fundamental nuclei, the dimension of which is . These fundamental nuclei are herein written at a layer level; such a choice has permitted to the author to treat both modelings which preserve the number of variables independent of the number of layers (Equivalent Single Layer Models, ESLM) and Layer Wise Models (LWM) in which the same variables are independent in each layer, at the same time. The variational statements and continuity requirements for stresses and displacements as well as non homogeneous boundary conditions at each interface, for displacement and/or transverse stress variables, are used to derive matrices from layers to multilayers and from elements to structure levels. A numerical investigation has been proposed to assess and compare the implemented finite elements. Numerical results and discussion have been provided in the extended version of this work. References
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