Third-order theory for the bending analysis of laminated thin and thick plates including the strain gradient effect

The aim of the paper is the development of a third-order theory for laminated composite plates that is able to accurately investigate their bending behavior in terms of displacements and stresses. The starting point is given by the corresponding Reddy’s Third-order Shear Deformation Theory (TSDT). This model is then generalized to consider simultaneously the Classical Laminated Plate Theory (CLPT), as well as the First-order Shear Deformation Theory (FSDT). The constitutive laws are modified according to the principles of the nonlocal strain gradient approach. The fundamental equations are solved analytically by means of the Navier methodology taking into account cross-ply and angle-ply lamination schemes. The numerical applications are presented to highlight the nonlocal effects on static behavior

Buckling analysis of three-phase CNT/polymer/fiber functionally graded orthotropic plates: Influence of the non-uniform distribution of the oriented fibers on the critical load

The current paper aims to analyze the influence on the critical buckling loads of the non-uniform distribution of the oriented fibers along the thickness direction of three-phase CNT/polymer/fiber functionally graded orthotropic plates. The various plies of the laminated plates are reinforced by both carbon nanotube (CNT) particles and conventional oriented straight fibers. The orthotropic features of such layers are provided by the reinforcing fibers which are functionally graded (FG) along the thickness coordinate. In the literature, CNTs represent generally the sole reinforcing phase and are assumed aligned and graded in the thickness direction. Here, instead, CNTs are randomly oriented and uniformly scattered in the matrix, whose properties are further improved by aligned, graded, straight and oriented fibers. A general power-law function is introduced to define the non-uniform features instead of the usual patterns presented in the literature (such as FG-X and FG-O), which can be included in the proposed approach as particular cases. The current methodology is tested through the comparison with the results available in the literature. The validation procedure is carried out for two-phases composites, considering also CNTs as straight and aligned reinforcing fibers, characterized by both uniform and graded properties. Several boundary conditions are also analyzed and verified. As proven by the numerical results illustrated in the paper, the variation of the through-the-thickness distribution of the fiber volume fraction is able to change noticeably the value of both uniaxial and biaxial critical buckling loads of arbitrarily restrained thin and thick plates. This effect should be considered in the manufacturing process and in the mechanical analysis of these structures

On the mapping procedure based on higher-order Hermite polynomials for laminated thin plates with arbitrary domains in gradient elasticity

The article presents a finite element mapping procedure for the coordinate change via higher-order derivatives to compute the fundamental matrices of laminated thin plates with arbitrary domains in gradient elasticity. In this context, the approximate solution requires Hermite interpolating functions. Therefore, conforming and nonconforming formulations are needed for both membrane and bending degrees of freedom, which require respectively (Formula presented.) and (Formula presented.) continuity. The aim of the current procedure is the possibility to remove the limitations related to the regular rectangular shape which typically characterizes this kind of elements and to introduce arbitrary distortions, discussing the influence of structured and unstructured meshes. The accuracy and convergence features of the methodology are presented through some numerical tests and compared to relevant literature

Higher-Order Weak Formulation for Arbitrarily Shaped Doubly-Curved Shells

The aim of this chapter is the development of an efficient and accurate higher-order formulation to solve the weak form of the governing equations that rule the mechanical behavior of doubly-curved shell structures made of composite materials, whose reference domain can be defined by arbitrary shapes. To this aim, a mapping procedure based on Non-Uniform Rational Basis Spline (NURBS) is introduced. It should be specified that the theoretical shell model is based on the Equivalent Single Layer (ESL) approach. In addition, the Generalized Integral Quadrature technique, that is a numerical tool which can guarantee high levels of accuracy with a low computational effort in the structural analysis of the considered shell elements, is introduced. The proposed technique is able to solve numerically the integrals by means of weighted sums of the values that a smooth function assumes in some discrete points placed within the reference domain

Time-dependent behavior of viscoelastic three-phase composite plates reinforced by Carbon nanotubes

none2noThe recent advancement in the manufacture and analysis of Carbon nanotubes (CNT) has persuaded many researchers to use them as the reinforcing phase of a polymer matrix. Nevertheless, it should be recalled that polymers show a viscoelastic behavior, so that their mechanical properties are functions of time due to the intrinsic nature of the material. In this paper, a Maxwell rheological model is employed to describe the time-dependency of the mechanical properties of the matrix in the framework of linear viscoelasticity. The viscoelastic matrix is enriched by both CNTs and oriented fibers to obtain the so-called three-phase composite materials. Such materials represent the main constituents of the plates investigated in this paper by means of the Reissner-Mindlin theory. The transient response, which is analyzed through the Newmark's scheme, is expressed in terms of central deflection and mechanical parameters for several configurations. The effect of the mass fraction of both reinforcing phases is discussed. The solutions are achieved numerically by means of a Finite Element (FE) code which implements the viscoelastic model and the proper homogenization technique in order to deal with three-phase composites.mixedBacciocchi, Michele; Tarantino, Angelo MarcelloBacciocchi, Michele; Tarantino, Angelo Marcell

Natural frequency Analysis of Functionally Graded Orthotropic Cross-Ply Plates Based on the Finite Element Method

This paper aims to present a finite element (FE) formulation for the study of the natural frequencies of functionally graded orthotropic laminated plates characterized by cross-ply layups. A nine-node Lagrange element is considered for this purpose. The main novelty of the research is the modelling of the reinforcing fibers of the orthotropic layers assuming a non-uniform distribution in the thickness direction. The Halpin−Tsai approach is employed to define the overall mechanical properties of the composite layers starting from the features of the two constituents (fiber and epoxy resin). Several functions are introduced to describe the dependency on the thickness coordinate of their volume fraction. The analyses are carried out in the theoretical framework provided by the first-order shear deformation theory (FSDT) for laminated thick plates. Nevertheless, the same approach is used to deal with the vibration analysis of thin plates, neglecting the shear stiffness of the structure. This objective is achieved by properly choosing the value of the shear correction factor, without any modification in the formulation. The results prove that the dynamic response of thin and thick plates, in terms of natural frequencies and mode shapes, is affected by the non-uniform placement of the fibers along the thickness direction

Finite anticlastic bending of hyperelastic laminated beams with a rubberlike core

A novel analytical approach to investigate the finite bending of hyperelastic laminated beams is presented. Two different nonlinear material models are taken into account, which are the compressible Mooney-Rivlin for rubberlike mediums and the Saint Venant-Kirchhoff for less deformable materials. The anticlastic bending is included in the formulation and the analytical expression of the transverse radius of curvature is presented. The stress analysis is performed in each layer separately, by considering the actual stored energy function of the constituents, in both Lagrangian and Eulerian frameworks. The finite bending of a sandwich beam is investigated in terms of stresses and stretches

Bending of hyperelastic beams made of transversely isotropic material in finite elasticity

The paper aims to investigate the finite bending of hyperelastic beams composed of transversely isotropic soft materials. The constitutive laws are obtained by including the transverse isotropy effects in the compressible Mooney-Rivlin model. A suitable expression for the stored energy function is introduced for this purpose, showing its dependency on five material invariants. A fully nonlinear three-dimensional beam model, including the anticlastic effect, is developed. The general analytical formulation allows to consider the influence of transverse isotropy on the Piola-Kirchhoff and Cauchy stress components, since it is presented in both Lagrangian and Eulerian frameworks. The validity of the current model is finally discussed. This study is justified by many innovative applications which require the use of transversely isotropic components, such as the finite bending of soft robots or biological systems

Finite bending of hyperelastic beams with transverse isotropy generated by longitudinal porosity

The paper deals with the finite bending analysis of transversely isotropic hyperelastic slender beams made of a neo-Hookean material with longitudinal voids. The fully nonlinear behavior of the structures is presented in the framework of three-dimensional finite elasticity. A semi-inverse approach is used to describe the beam kinematics, which includes the anticlastic effect. The theoretical framework is developed in both Lagrangian and Eulerian reference systems. Explicit formulas are obtained for stretches and stresses, in a general framework valid for transversely isotropic beams. The effect of porosity on the Piola-Kirchhoff and Cauchy stress components is then discussed. The results are all obtained and validated analytically, and could be helpful to model structural systems in the fields of bioengineering and soft-robotics which exhibit both large displacements and deformations