A solid-shell layerwise finite element for non-linear geometric and material analysis R.A.S. Moreira * , R.J. Alves de Sousa, R.A.F. Valente Departamento de Engenharia Mecânica, Universidade de Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal article info Article history: Available online 25 October 2009 Keywords: Sandwich structures Enhanced assumed strain Solid-shell Multilayer model abstract The application of layerwise theories to correctly model the displacement field of sandwich structures or laminates with high modulus ratios usually employs plate or facet-shell finite element formulations to compute the element stiffness and mass matrices for each layer. In this work an alternative approach is proposed, using a high performance hexahedral finite element to represent the individual layer mass and stiffness. This eight-node hexahedral finite element is formulated based on the application of the enhanced assumed strain method (EAS) to solve several locking pathologies coming from the high aspect ratio of the finite element and the usual incompressibility condition of the core materials. The solid-shell finite element formulation is introduced in the layerwise theory through the definition of a projection operator, based on the finite element variables transformation matrix. The non-linear geometric and material capabilities are introduced into the finite element formulation, allowing for the representation of large displacements, large deformation and material non-linear behaviors. The developed formulation is numerically tested and benchmarked, being validated by using published experimental results obtained from sandwich specimens. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Sandwich structures with light cores, using honeycomb pat- terns, viscoelastic layers or polymeric foams between two stiff skin plates, are attractive solutions for structural applications, provid- ing an interesting stiffness/weight ratio and, when properly de- signed for passive dynamic control applications [1], leading to internal material damping capacity. Sandwich materials can be re- garded as a laminated structure which, in opposition to the lami- nated cross-ply and angle-ply composites, is divided into skins and core layers. The skins are usually made of stiff materials, whereas the core is properly tailored to achieve a set of specific properties, being usually made from light materials or structures – to rise the assembly inertia moment with a reduced weight in- crease – high damping polymers – to improve the passive dynamic control capacity – or even active materials for active dynamic con- trol purposes. In order to spatially model layered structures, different ap- proaches may be used. Equivalent Single Layer (ESL) models, like the First-order Shear Deformation Theory (FSDT) or the Higher-or- der Shear Deformation Theory (HSDT) [2–5] can be easily imple- mented and the required number of degrees-of-freedom does not depend upon the number of layers. Despite the higher-order description of the displacement field, even the HSDT model may not be able to correctly describe the displacement field distribution along the thickness when adjacent layers with significant modulus ratios are present in the laminate. In those cases, zig-zag [6] or lay- erwise models [7] can provide valuable solutions for such requirement. Sandwich structures with soft cores, like those achieved when inserting thin viscoelastic layers or foams between two stiff skin plates for damping purposes, are usually not straightforward to simulate due to the difficulties related to the spatial model of the layered assembly, which must be able to accurately represent the high shear pattern of the core and, therefore, the related damping mechanism, or simply to correctly represent the through-the- thickness deformation of the core. The usual approach, by means of a layered assembly of plate and solid finite elements with nodal linkage or rigid link elements [8–10], provides an expeditious mod- eling approach, which can be straightforwardly implemented within a commercial finite element package. However, such ap- proach can suffer from numerical deficiencies, such as volumetric and transverse shear locking phenomena [10] and, even if special care is taken to overcome the numerical locking onset, such ap- proach may lead to a cumbersome and time-consuming modeling task, particularly to model three-dimensional shell-type struc- tures. Additionally, such modeling approach requires the modifica- tion of the entire finite element mesh for any needed reconfiguration of the laminate structure, which is obviously not a straightforward methodology during design and optimization simulation tasks [11]. 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.10.032 * Corresponding author. E-mail addresses: rmoreira@ua.pt (R.A.S. Moreira), rsousa@ua.pt (R.J. Alves de Sousa), robertt@ua.pt (R.A.F. Valente). URL: http://www.mec.ua.pt (R.A.S. Moreira). Composite Structures 92 (2010) 1517–1523 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct