scholarly journals A Stress Approach Model of Functionally Graded Shells

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Carlos Humberto Rubio Rascón ◽  
Alberto Díaz Díaz ◽  
Axel Fernando Domínguez Alvarado
Keyword(s):  
Finite Element ◽  
Stress Field ◽  
Functionally Graded Material ◽  
Degrees Of Freedom ◽  
Computational Cost ◽  
Good Alternative ◽  
Functionally Graded ◽  
Equilibrium Equations ◽  
Finite Element Calculations ◽  
Graded Material

This paper presents a model called SAM-FG (stress approach model of functionally graded shells) for linear elastic, thin, and moderately thick shells made of functionally graded materials. The model is an extension of the SAM-H model, originally created for homogeneous shells. Assuming that the material is orthotropic and that one of its orthotropic directions is the thickness direction, the extension consists in considering that the 3D compliance tensor may depend on the thickness coordinate. The model starts with a tunable polynomial approximation of the 3D stress field that contains the same generalized forces as SAM-H. This stress approximation verifies the 3D equilibrium equations and the stress boundary conditions at the faces of the shell. As in SAM-H, 5 generalized displacements appear in SAM-FG. By applying the Hellinger–Reissner functional and Reissner’s variational method, the generalized forces, strains, and equations in SAM-FG turn out to be the same as in SAM-H, except for the generalized constitutive equations. To prove the accuracy of the model, SAM-FG is first applied to a simply supported, functionally graded plate and its results are compared to other models. To validate the model for shell-like structures, SAM-FG results are compared to those obtained with solid finite element calculations for three case studies of structures subjected to an internal pressure. The first one deals with a hollow sphere made of an isotropic functionally graded material. The second case considers a hollow cylinder made of an orthotropic functionally graded material. In the last case, a catenoid with an isotropic functionally graded material is studied. In all cases, the mean displacements are correctly predicted, even if the main purpose of the SAM-FG model is not to calculate these fields accurately. The stress field approximations are very accurate, and since the implementation of the shell model in a finite element code would imply 5 degrees of freedom per node, SAM-FG is a good alternative to solid finite element calculations for the structural analysis of functionally graded shells with a reasonable computational cost.

scholarly journals Extended finite -element method for modeling the mechanical behavior of functionally graded material plates with multiple random inclusions

2017 ◽  
Vol 20 (K3) ◽  
pp. 119-125
Author(s):  
Bang Kim Tran ◽  
Huy The Tran ◽  
Tinh Quoc Bui ◽  
Thien Tich Truong
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Level Set Method ◽  
Level Set ◽  
Functionally Graded Material ◽  
Extended Finite Element Method ◽  
Functionally Graded ◽  
Extended Finite Element ◽  
Graded Material ◽  
Element Method

Functionally graded material is of great importance in many engineering problems. Here the effect of multiple random inclusions in functionally graded material (FGM) is investigated in this paper. Since the geometry of entire model becomes complicated when many inclusions with different sizes appearing in the body, a methodology to model those inclusions without meshing the internal boundaries is proposed. The numerical method couples the level set method to the extended finite-element method (X-FEM). In the X-FEM, the finite-element approximation is enriched by additional functions through the notion of partition of unity. The level set method is used for representing the location of random inclusions. Numerical examples are presented to demonstrate the accuracy and potential of this technique. The obtained results are compared with available refered results and COMSOL, the finite element method software.

Задачи о дисперсии волн в неоднородных волноводах: методы решения (обзор). Часть I

2021 ◽  
Vol 27 (2) ◽  
pp. 227-260
Author(s):  
С. И. Жаворонок ◽  
Keyword(s):  
Functionally Graded Materials ◽  
Functionally Graded Material ◽  
Matrix Method ◽  
Degrees Of Freedom ◽  
Normal Wave ◽  
Wave Dispersion ◽  
Functionally Graded ◽  
Graded Materials ◽  
Generalized Fourier Series ◽  
Graded Material

A brief review of the modern state-of-the art and tendencies of further development of various methods of solution of wave dispersion problems in heterogeneous functionally graded elastic waveguides is presented. Main types of functionally graded materials and structures, including gradient thon-walled structures, and their main engineering applications is discussed. The main difficulties of modelling of the stress-strain state of functionally graded shells and plates are pointed, as well as the possible ways to overcome such difficulties. The main theoretical bases of definition of effective constitutive constants of functionally graded materials and their possible estimates used in the practice are considered. Main dependencies of the effective constitutive constants of a functionally graded material on coordinates used for the mathematical modelling of the dynamics are also shown. The statement of the dynamics problem for a functionally graded waveguide and the appropriate statement of the normal wave dispersion problem are pointed. The presented Part I of the review consider some analytical methods of solution of dispersion problems, mainly the matrix ones based on the formulation of the steady dynamics problem in the image space as a first-order ordinary differential equations system. The state vectors corresponding to the useful Cauchy and Stroh formalisms are introduced, and the appropriate governing equations and the boundary conditions on waveguide’s faces are presented. Classical methods for solving the steady dynamics problem for a laminated waveguide are briefly described, which could be a basis for the further approximation of a functionally graded material by a system of layers with constant properties, i.e. the transfer matrix method, its main modifications developed to ensure the stability of calculations, and the global matrix method. Then, the intensively developed last 15 years reverberation matrix method, stiffness matrix method, and the Peano series method are discussed. Some key solutions of the wave dispersion problems for heterogeneous layers are presented; such solutions improve the efficiency of approximation of a functionally graded structure by a laminated one. The implicit solution of the general problem of steady dynamics for a waveguide with arbitrary gradation law is shown. The key features of the discussed matrix methods are pointed briefly as well as their main drawbacks. In the Part II, the main attention will be paid to methods of semi-analytical solution of dispersion problems based on the approximation of a waveguide by an equivalent system with a finite number of degrees of freedom: power series, generalized Fourier series, semi-analytical finite elements. spectral elements, as well as methods based on various theories of plates and shells.

NONLINEAR MECHANICAL BENDING OF FUNCTIONALLY GRADED MATERIAL PLATES UNDER TRANSVERSE LOADS WITH VARIOUS BOUNDARY CONDITIONS

2011 ◽  
Vol 02 (02) ◽  
pp. 237-258 ◽  
Author(s):  
MOHAMMAD TALHA ◽  
B. N. SINGH
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Functionally Graded Material ◽  
Functionally Graded ◽  
Nonlinear Strain ◽  
Various Boundary Conditions ◽  
Transverse Loads ◽  
Nonlinear Mechanical ◽  
Graded Material ◽  
Mechanical Bending

Nonlinear mechanical bending of functionally graded material (FGM) plates under transverse loads with various boundary conditions are presented. The material properties of the FGM plates are graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The theoretical nonlinear finite element formulations are based on the higher-order shear deformation theory, with a special modification in the transverse displacement in order to estimate the parabolic distribution of transverse shear strains through the plate thickness. The Green–Lagrange nonlinear strain–displacement relation with all higher-order nonlinear strain terms is included to account for the large deflection response of the plate. The fundamental equations for FGM plates with traction-free boundary conditions on the top and bottom faces of the plate are accomplished using variational approach. Results have been achieved using a C0 continuous isoparametric Lagrangian finite element with 13 degrees of freedom per node. Convergence and comparison studies have been performed to ascertain the effectiveness of the present model. Numerical results are highlighted for different thickness ratios, aspect ratios, and role played by the constituent volume fraction index with different boundary conditions.

Effects of piezoelectric rings on electro-elasto-dynamic behaviour of functionally graded cylindrical shell

2016 ◽  
Vol 28 (2) ◽  
pp. 272-289 ◽  
Author(s):  
Mohammadreza Saviz
Keyword(s):  
Finite Element ◽  
Cylindrical Shell ◽  
Functionally Graded Material ◽  
Volume Fraction ◽  
Virtual Work ◽  
Electrical Potential ◽  
Axial Direction ◽  
Functionally Graded ◽  
Radial Coordinate ◽  
Graded Material

A layer-wise finite element approach is adopted to analyse the hollow cylindrical shell made of functionally graded material with piezoelectric rings as sensor/actuator, under dynamic load. The mechanical properties of the substrate are regulated by volume fraction as a function of radial coordinate. The thickness of functionally graded material shell and piezo-rings is divided into mathematical sub-layers and then the general layer-wise laminate theory is formulated through introducing piecewise continuous approximations across the thickness, accounting for any discontinuity in derivatives of the displacement at the interface between the ring and cylinder. The virtual work statement including structural and electrical potential energies yields the three-dimensional governing equations which are reduced to two-dimensional differential equations, using layer-wise method. For axisymmetric case, the resulted equations are solved with one-dimensional finite element method in the axial direction. By assembling stiffness and mass matrices, the required stress and displacement continuities at each interface and between the two adjacent elements are forced. The results for free vibration and static loading are applied to study the convergence and verified by comparing them to solutions of similar existing problems. The induced deformation by piezoelectric actuators as well as the effect of rings on functionally graded material shell is investigated.

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