Nonlinear vibrations of symmetric cross-ply laminates via thermomechanically coupled reduced order models

Thermomechanically coupled, geometrically nonlinear, laminated plates are addressed through a unified 2D formulation, by considering classical and third-order shear-deformable von Karman models, along with correspondingly consistent linear and cubic variations of the temperature along the thickness. Minimal dimension reduction of the mechanical problem is accomplished for symmetric cross-ply laminates, ending up for both models to a coupled three-mode reduced model with terms and coefficients of variable nature depending on the variety of mechanical and/or thermal excitations. Nonlinear vibrations of the classical model are investigated in conditions of thermal dynamics either passively entrained by the harmonically varying transverse load via the existing coupling terms, or also playing some active role owed to a temperature difference with respect to the surrounding medium

Modelling and Nonlinear Dynamics of Third-Order Thermomechanically Coupled Laminated Plates

Thermomechanically coupled, geometrically nonlinear, laminated plates are addressed through a unified 2D formulation that integrates mechanical and thermal aspects and consistently accounts for cubic variations along the thickness of both in-plane displacement components and temperature. It allows to address a variety of thermal boundary conditions on the plate upper and lower surfaces. Minimal dimension reduction of the problem is pursued for symmetric cross-ply laminates. A numerical case study provides hints on the potential of the reduced model for the analysis of thermomechanical coupling effects on the system nonlinear dynamics

Modeling, Dimension Reduction, and Nonlinear Vibrations of Thermomechanically Coupled Laminated Plates☆

Abstract A unified formulation of thermomechanical, geometrically nonlinear, laminated plates that integrates mechanical and thermal aspects is presented. It allows for constructing and comparing a variety of continuous models of different mechanical richness and with full thermoelastic coupling embedded, as well as for deriving minimal reduced order models suitable to provide useful information on fundamental thermomechanical phenomena occurring in the system nonlinear and complex dynamics. Comparative numerical investigations of free and forced vibrations can be carried out through both models of three, fully coupled, ordinary differential equations and simplified, partially coupled, models of two, or even one, ODEs, with the aim to unveil the actual importance of accounting for the various terms to reliably describe the most important thermomechanical effects on the system response

3D limit analysis of masonry pavilion domes on octagonal drum subjected to vertical loads

Within the framework of limit design applied to masonry structures, this paper aims at analyzing the different behavior of a pavilion dome according to the adopted construction and reinforcement technologies. By using the static theorem applied to the dome discretized in rigid macro-blocks of variable shape aligned along parallels and meridians, a mathematical model have been constructed in order to search for the load collapse multiplier, and thus to evaluate the degree of structural safety. Then, the associated failure mechanism is represented at the instant in which the collapse is reached. The program that implement the modeling is sufficiently versatile and, in addition to the mechanical characteristics, allows to define the intrados profile, the thickness variability, as well as to insert any window opening in the drum, the lantern at the top and the hoops at each level. The results shown here concern some numerical applications carried out on a theoretical dome, as well as those related to a first approach to the analysis of the dome of Santa Maria del Fiore in Florence by Brunelleschi

Unified 2D continuous and reduced order modeling of thermomechanically coupled laminated plate for nonlinear vibrations

A unified formulation of the thermomechanical problem of laminated plates with von Karman nonlinearities, undergoing finite amplitude vibrations, is presented. It integrates mechanical and thermal aspects, by addressing them in parallel via the introduction of generalized 2D variables and equations also for the latter. The formulation virtually embeds a multitude of possible models, resulting from different assumptions about the plate mechanical and thermal configurations. The obtained continuous model is then subjected to a minimum reduction via Galerkin procedure. Some analyses of free and forced nonlinear vibrations under variable mechanical and/or thermal excitations are also carried out, to get some hints on the importance of different thermal aspects associated with membrane and bending dynamics, and on the possibility to catch them via variably simplified models

Continuous modeling and minimal dimension reduction for the nonlinear dynamics of thermoelastic laminated plates

In laminated plates, different approximations for the deformation of the reference plane and the shear-warping of the cross section result in distinct geometrically nonlinear and shear deformation models. In addition to these mechanical features, it is often important to evaluate the effects of thermal phenomena, by properly selecting some relevant simplifying assumptions. When formulating ROMs to be used for highlighting features of nonlinear dynamic response, shear deformations are usually taken into account, whereas nonlinear curvatures and thermo-elastic coupling are generally overlooked. In this work, modeling of thermoelastic nonlinear laminated plates is addressed in the background of classical Tonti diagram for physical theories [1]. It allows a comprehensive and unified classification/comparison of various assumptions concerning (i) geometric nonlinearities, (ii) shear deformability and (iii) thermal involvement. From a mathematical point of view, the resulting continuous mo

Shear deformable composite plates with nonlinear curvatures: modeling and nonlinear vibrations of symmetric laminates

Moving from a general plate theory, a modified general shear deformable laminated plate theory (MGFSDT) exhibiting nonlinear curvatures but still allowing for some worth features of linear curvature models (von Karman) is formulated. Starting from MGFSDT partial differential equations, a minimal discretized model (Duffing equation) for symmetric cross-ply laminates nonlinear vibrations, whose coefficients account for shear deformability and nonlinear curvatures, is obtained via the Galerkin procedure. The variable features of such a Duffing model as obtainable via alternative kinematic assumptions at the continuum level are highlighted. Through the comparison of a number of underlying models in different technical situations, information on the influence of shear deformability on system nonlinear response and on the influence of nonlinear curvatures are obtained. Frequency-response curves through a multiple scale analysis are presented for different continuous models, kinds of material, mode numbers and boundary conditions

Nonlinear curvature-based model and resonant finite-amplitude vibrations of symmetric cross-ply laminates

Moving from a general plate theory, a modified general classical laminated plate theory (MGCLPT) exhibiting nonlinear curvatures but still allowing for some worth features of linear curvature models (von Karman) is formulated. Starting from MGCLPT partial differential equations, a minimal discretized model suitable for the analysis of resonant finite-amplitude vibrations of symmetric cross-ply laminates, with immovable or movable supports, is obtained via the Galerkin procedure. Periodic responses of a single-mode model and of a 3:1 internally resonant two-mode model excited at primary resonance are obtained via the multiple time scale method. The influence of various system parameters (thickness ratio, plate aspect, number of laminae, kind of material, mode number) is addressed, and the comparison of nonlinear vibration results as obtained with the MGCLPT and the von Karman models for different boundary conditions shows some interesting differences. © 2012 Elsevier Ltd. All rights reserved