Keynote Lectures

Layered Structures are used in many applications because of the possibility to design materials with prescribed properties. At the same time, they are often thin and one gets a lightweight structure. During the last 100 years, many suggestions were made to improve the Euler-Bernoulli beam theory or the Kirchhoff plate theory. The idea was to have simple one- or two-dimensional theories, but better predictions of the mechanical behavior of such structures which is necessary for high inhomogeneous in the thickness direction structures. The mechanical behavior is influenced by both the mechanical properties’ ratios and the thickness size of the different layers. There are four main directions in development beam or plate theories: - Assuming hypotheses with respect to the stress, strain, etc. states one can deduce the full set of governing equations including the constitutive relations. - Introducing a deformable surface a set of two-dimensional continuum mechanics equations can be established. The problem is the identification of the constitutive parameters from the three-dimensional images. - Assuming a small parameter, mathematical techniques can be used to formulate the two-dimensional equations. Such techniques are series expansions or asymptotic integration. - Consistent formulation are based on power series expansions and on consistent truncation of the series the strains, etc. It is obvious that each approach has advantages and disadvantages. Anyway, on gets only an approximation of the structures, which are three-dimensional even if the thickness is small. The last one is also the reason for inaccuracies of Finite Element based simulations. On different aspects will be reported in the lecture.

The effective nonlinear exact geometry or geometrically exact (GeX) four-node laminated composite and piezoelectric solid-shell elements using the method of sampling surfaces (SaS) are presented. The term GeX reflects the fact that the parametrization of the middle surface is known and, therefore, the coefficients of the first and second fundamental forms and Christoffel symbols are taken exactly at element nodes. The SaS formulation is based on the choice inside the layers of an arbitrarily number of SaS parallel to the middle surface and located at Chebyshev polynomial nodes to introduce the displacements and electric potentials of these surfaces as basic shell unknowns. The outer surfaces and interfaces are also included into a set of SaS. The SaS shell formulation uses the objective Green-Lagrange strain tensor that exactly represents large rigid-body motions of the shell in a curvilinear coordinate system. The developed solid-shell elements are based on the hybrid-mixed method through the Hu-Washizu variational formulation. The tangent stiffness matrices are evaluated by efficient 3D analytical integration proposed by the author. As a result, both GeX solid-shell elements exhibit a superior performance in the case of course meshes and allow the use of load increments, which are much larger than possible with existing displacement-based finite elements. They can be utilized for the analysis of the second Piola-Kirchhoff stresses and electric displacements in thick and thin laminated composite and piezoelectric doubly-curved shells because the SaS solutions asymptotically approach the exact solutions of elasticity and piezoelectricity as a number of SaS tends to infinity.

The complexity of composite structures (laminates, sandwich panels) calls for structural models that go beyond the limitations of classical theories such as CLT, based on Kirchhoff-Love’s hypotheses, and FSDT, based on Reissner-Mindlin’s hypotheses. A huge number of models have been thus developed in the past decades to include refined descriptions of the kinematics as well as the stress response, as well as towards meeting the so-called C0z-Requirements that should be met for taking into account the mesoscale heterogeneity of a composite stack. The large variety of models that have been proposed, ranging from Equivalent Single Layer (ESL) to Layer-Wise (LW) and Zig-Zag models, can be explained by the fact that their accuracy is strongly application-dependent. Variable kinematics approaches offer a generic framework for the mechanics of structures, which can be exploited for assessing the accuracy of such axiomatically formulated models and, eventually, for identifying the ``best’’ model in terms of the ratio between accuracy and computational cost. The Sublaminate Generalized Unified Formulation (SGUF) is the formal extension of the well established Carrera’s Unified Formulation (CUF) to formulate arbitrary models for groups of plies (the sublaminates): in SGUF, different axiomatic models (ESL, ZigZag, LW) are introduced in each sublaminate by referring to Demasi’s Generalized Unified Formulation (GUF), and the model of the whole composite stack is obtained as a LW assembly of sublaminates. Displacement based plate models expressed in SGUF have been implemented within a highly efficient Ritz-type solution as well as within a dedicated, locking-free FEM. The talk provides an overview of this approach along with numerical applications highlighting its current capabilities. Linear elastic problems are addressed to highlight the robust FEM implementation, which is also available as Abaqus User Element. The Ritz-type solution is used to discuss the extension of the SGUF approach to geometrically nonlinear bending problems. Applications to heterogeneous composites are also considered, in particular by addressing the modal and harmonic response of composite panels hosting piezoelectric plies and frequency-dependent viscoelastic materials.

Developing of conventional continuum-based numerical models in composite materials has been grown continuously over the past decades. Diversity of these models, however, prevents them from the practical applications. Machine Learning (ML) based methods, in contrast, have a great potential to construct generalized surrogate models to realize the behavior of composite materials from the available data under various conditions and loading terms. This yields particularly to reliable results for any nonlinearity between input and outputs. Assuming the availability of limited data on the response of composite materials, ML methods employ the deep learning algorithms constructed based on the multi-layer artificial neural networks (ANN) to build the input-output relationships. The data can be extracted from the numerical simulations or experimental data. In this contribution, various aspects of the ML based modeling for composite materials including training and testing methods are investigated. Particularly, the proper orthogonal decomposition (POD) and the dynamic mode decomposition (DMD) in combination with the ANN is introduced to simulate the dynamical behavior of composite materials under material uncertainties in load terms and modeling parameters. The method leads to a drastically reduction in the rank of system matrices to improve the computational time. Application of the method is presented for the dynamic analysis of fiber reinforced composite plates under material uncertainties.

Automated Fiber Placement (AFP) technology has revolutionized composites manufacturing and is quickly replacing traditional manual production. Among others, the notable advantage of AFP is that the designer is no more restricted to the use straight fibers, which means it is possible to have steered fiber paths designed for optimizing structural performance. The scope of this work is to identify the manufacturing parameters and associated constraints imposed by these parameters on the mathematical design space and introduce this in an optimization framework to design future aerospace structures (wings, fuselages etc.) for superior mechanical performance. In this talk, the design of a structural panel with a cutout to minizime stress concentration and the design of a skin to maximize buckling loads will be addressed. Furthermore, the effects of manufacturing induced unintended defects on the strength (tensile and compressive) of these strucures will also be addressed through experimentally validated computational models that account for the material microstructure.