Paolo Vannucci

Stiffness design of laminates using the polar method

This paper is devoted to the analysis of elastic properties of anisotropic laminas using the so-called polar representation method: this is an effective mathematical tool to analyse two-dimensional elastic problems. By this method, the authors have been able to find a particular class of solutions to some special inverse problems concerning laminates made by anisotropic layers. The properties of these solutions are described and discussed, along with some general results.

A special class of uncoupled and quasi-homogeneous laminates

Abstract This paper deals with two main problems in laminate design: the search for uncoupled and quasi-homogeneous laminates. Using the polar representation method, the authors show the existence of a particular class of mathematically exact solutions to these two problems. An important feature of these solutions is that they are independent of the orientations of the layers. In fact, these orientations are not fixed by the method, and each solution determines in reality only a stacking sequence, where each layer belongs to a group of plies having the same orientation. The orientations remain undetermined, and it is up to the designer to fix them. In any event, whether the laminate is uncoupled or quasi-homogeneous, the orientations of the layers will reamin free, and this is a true advantage for an optimisation procedure when supplementary conditions are required The characteristics of the solutions and the general results found by the authors are discussed in the paper, which concludes with some numerical examples.

A Two-Level Procedure for the Global Optimum Design of Composite Modular Structures—Application to the Design of an Aircraft Wing

This work concerns a two-level procedure for the global optimum design of composite modular structures. The case-study considered is the least weight design of a stiffened wing-box for an aircraft structure. The method is based on the use of the polar formalism and on a genetic algorithm. In the first level of the procedure, the optimal structure is designed as it was composed by a single equivalent layer, while a laminate realizing the optimal structure is found in the second level. The method is able to automatically find the optimal number of modules, no simplifying assumptions are used, and it can be easily generalized to other problems. The work is divided into two parts: the theoretical formulation in this first part, the genetic procedure and some numerical examples in the second one.

A Two-Level Procedure for the Global Optimum Design of Composite Modular Structures—Application to the Design of an Aircraft Wing: Part 2: Numerical Aspects and Examples

This work concerns a two-level procedure for the global optimum design of composite modular structures. The case-study considered is the least weight design of a stiffened wing-box for an aircraft structure. The method is based on the use of the polar formalism and on a genetic algorithm. In the first level of the procedure, the optimal structure is designed as composed by a single equivalent layer, while a laminate realizing the optimal structure is found in the second level. The method is able to automatically find the optimal number of modules; no simplifying assumptions are used and it can be easily generalized to other problems. The work is divided into two parts: the theoretical formulation in the first part, the genetic procedure and some numerical examples in this second one.

A special planar orthotropic material

This paper shows the existence of a particular type of planar orthotropic material, here denoted for the sake of brevity as R0-orthotropic. The number of independent elastic constants for these materials is three, and not four as for a general orthotropic layer, but these constants have only two orthogonal axes of symmetry. The way to obtain a R0-orthotropic layer is discussed in the paper, along with the advantages in its use.

A new method for generating fully isotropic laminates

In this paper the authors propose some new kinds of isotropic laminates, made with identical anisotropic layers. In particular, these laminates satisfy some conditions which generalise the well-known Werren and Norris rules, in order to obtain fully isotropy, that is, isotropy of the three tensors A, B and D. To this purpose, the authors utilise some results found in a preceding research, namely the so-called quasi-trivial solutions. The way to form particular isotropic laminates that do not follow the Werren and Norris rule is also indicated. The paper ends with some numerical examples which illustrate the theoretical results found.

Anisotropy and symmetry for elastic properties of laminates reinforced by balanced fabrics

In this article, we present a theoretical study on elastic properties of laminates composed by balanced fabric layers. Using the polar representation method for plane elastic tensors, we first describe some properties of symmetry of a general laminate composed by balanced fabrics and we write the formulas expressing positions of its axes of symmetry. Then, limiting our study to laminates composed of identical plies, we solve two problems of symmetry of the laminate elastic tensors: uncoupling and quasi-homogeneity. We found all the solutions of the uncoupling problem for the case of 3-, 4- and 5-ply laminate and all those of the quasi-homogeneity problem for the case of 4-, 5- and 6-ply laminate: it is worth noting that this class of solutions was not known before.

Optimization of Laminated Composites by Using Genetic Algorithm and the Polar Description of Plane Anisotropy

The object of the present work is the development and application of a quite general approach to optimal design of composite laminates where elastic symmetries can also be explicitly expressed as criteria of the optimization process. Our formulation is in the form of a highly nonlinear and nonconvex single- or multi-objective optimization problem subject to equality and inequality constraints. We show here applications to the design of maximum stiffness, maximum buckling load, maximum eigenfrequencies, maximum strength for laminated plates, as well as combinations of the aforementioned criteria; all types of elastic symmetries can be taken into account. We show here a number of numerical solutions found using the genetic algorithm BIANCA.

A two-level procedure for the global optimization of the damping behavior of composite laminated plates with elastomer patches

This work concerns a two-level procedure for the global optimum design of hybrid elastomer/composite modular structures. The goal of the procedure is the maximization of the first Nf modal loss factors of the structure, satisfying mechanical constraints on the weight and on the bending stiffness, feasibility constraints on the admissible moduli for the constitutive laminates, along with geometric constraints on the positions of the viscoelastic patches. At the first level of the procedure, the optimization of the damping behavior of the structure is carried out: the optimization variables at this stage are the number of elastomer patches (modules), as well as their geometrical parameters (position, thickness and diameter), along with the material and geometric parameters of the composite laminated structure (elastic moduli, thickness of the laminate). The composite structure supporting the elastomer patches is thus optimized using a free-material approach, via the polar representation of 2D elasticity, and the second level of the optimization consists in finding the laminate stacking sequence satisfying the optimal elastic moduli and thickness issued from the first step. The method is able to automatically determine the optimal number of modules and it does not need the introduction of any simplifying assumption. The proposed approach relies on one hand, on the application of the well-known Iterative Modal Strain Energy (IMSE) method for the evaluation of the dynamic response of the structure, and on the other hand on the use of the polar formalism for the representation of the elastic anisotropic behavior of composite laminates as well as of a genetic algorithm as optimization tool to perform the solution search. We will illustrate the application of our approach to the optimization of the damping behavior of a rectangular composite plate with a discontinuous aperiodic distribution of viscoelastic material. The numerical results show the effectiveness of the proposed strategy.

On bending-tension coupling of laminates

This paper deals with the evaluation of coupling between the in and out of plane behaviours of a laminate composed of identical anisotropic layers. The subject has been motivated by problems arising from coupling in modern practical industrial applications. Coupling is an effect which is often undesired by laminate designers, and some rules and solutions, recalled in the paper, are known today to counter it. Two subjects are analysed in the article: firstly, it is considered the problem of finding the stacking sequence angles which determine the maximum coupling of a laminate whose number of plies and elementary layer properties are known. Secondly, the sensitivity of uncoupling to an orientation defect is analysed, that is, how coupling changes when an error in the orientation of one layer is made: a simple and exact formula assessing the degree of coupling, which is the ratio of the real coupling with respect to the possible maximum, is given. The polar method, proposed by G. Verchery, has been used to develop the research, which is made in the theoretical framework of the Classical Laminated Plate Theory.