Graded Finite Element Research Articles

FINITE ELEMENT MODELLING AND ANALYSIS OF FREE VIBRATIONS OF AXIALLY FUNCTIONALY GRADED BEAMS WITH NON-UNIFORM CROSS-SECTIONS

The finite element package Abaqus is used for modeling and analysis of free vibrations of axially functional graded beams with non-uniform cross-sections. Considerable attention has been paid to this aspect of the geometry, as it has a significant effect on the behavior of the beam as a structural component. User-defined material model subroutines (UMAT) were developed in a unified way for one- and three-dimensional models of the specified beams to provide numerical implementation of functional material gradients within each finite element. UMAT procedures were programmed in FORTRAN in the MS Visual Studio environment and compiled with the main program package using the Intel Fortran Compiler. In the procedures, material heterogeneity was assigned to each material integration point of appropriate standard one- and three-dimensional finite elements, eliminating the need to design graded finite elements via user-defined element subroutine. Frequencies and mode shapes of free vibrations of one- and three-dimensional axially functional graded beams with non-uniform cross-sections were found using the Lanczos method in Abaqus. An analysis of the accuracy and effectiveness of the proposed modeling approach was carried out, comparing the obtained results with data known in the literature. The presented modeling technique based on user-defined material subroutines provides valuable opportunities for scientists and engineers engaged in the dynamic analysis of structural components made of functionally gradient materials and having variable geometry along the axial direction. Keywords: axially functionally graded beams; non-uniform cross-section beams; free vibrations; graded finite element; Abaqus user defined subroutines

Non-conforming and conforming five-node quadrilateral graded finite elements

This study evaluates the performance of conforming and non-conforming transition graded finite elements dealing with graded materials. Two-dimensional 5-node non-conforming quadrilateral element (Q5) is used to transition from one large 4-node linear quadrilateral to two small 4-node linear quadrilateral elements (Q4). In addition, a conforming Q5 element is used in between one Q4 and one 8-node quadratic elements (Q8). A single element test of Q5 graded element is done. Then, the accuracy and convergence study of Q5 graded element are examined. We found that the use of Q5 transition elements improved computational efficiency while maintaining the accuracy of the solutions.

Torsional buckling response of FG porous thick truncated conical shell panels reinforced by GPLs supporting on Winkler elastic foundation

In the present research, torsional buckling response of thick truncated conical shell panels supporting on Winkler-type elastic medium has been surveyed. The shell is constructed from a porous metal material reinforced by graphene platelets (GPLs) in which the porosity and volume weight fraction of nanofillers are graded along the thickness of shell. Three functions for porosity distribution and five patterns of GPLs are examined. To estimate the Young modulus of the shell, Tsai-Halpin micromechanical model, and for its mass density, extended rule of mixture is employed. Linear theory of 3D elasticity in conjunction with the virtual work principle and numerical graded finite element method (FEM) are applied to derive the state of equilibrium in pre-buckling mode. Torsional buckling forces are derived by applying nonlinear Green strains and by deriving geometric stiffness matrix of the system. The effect of various parameters such as coefficient of porosity, various distributions of porosity and different nanofiller patterns, weight fraction of graphene nanofillers, semi vertex angle of cone, span angle, stiffness of elastic medium and various boundary conditions on torsional buckling loads of functionally graded (FG) porous truncated conical panel reinforced by GPLs have been presented. The main purpose of this research is obtaining the best distribution of porosity and GPLs pattern and investigating the influence of adding nano particles on the torsional buckling forces of the shell. Results denote that employing GPL-X pattern in conjunction with porosity distribution 1 provides the maximum value of buckling loads. Besides by adding the nano particles, the amount of buckling loads will increase 100%.

Incompatible Graded Finite Elements for Orthotropic Nonhomogeneous Media

Orthotropic materials or composites are challenging to model due to different elastic properties in two in-plane directions, which requires sufficient mesh refinement to attain desired accuracy. In this regard, an accurate and efficient incompatible graded element is developed for modeling orthotropic functionally graded materials. Properties of Graded finite elements such as Young’s moduli (E11, E22), shear modulus (G12) and Poisson’s ratio (v12) vary. A new incompatible graded element is developed using user subroutines in Abaqus and are compared to available exact solutions. In this study, performance (QM6) graded element is compared with lower-order and higher-order compatible elements (Q4 and Q8) as well as linear triangular (T3) and quadratic triangular (T6) elements with aid of several numerical examples. Additionally, orthotropic graded plate with carbon fibers properties as well as a circular disc with orthotropic properties under internal pressure is included. Furthermore, a curved beam with radially graded properties under the bending moment is studied. Additionally, the performance of various graded elements is compared with analytical solutions using Russell error.

Evaluation of the mechanical properties of graphene-based nanocomposites incorporating a graded interphase based on isoparametric graded finite element model

ABSTRACT The effective mechanical properties of graphene-based nanocomposites considering a graded interphase are evaluated by using nanoscale sandwich representative volume element (RVE) and embedded RVE based on finite element method (FEM). Both sandwich RVE and embedded RVE are considered as a three-phase composite structure, in which the material properties of interphase are represented as a linear or an exponential function of the thickness direction coordinate z. The FEM is used to model the RVEs in order to evaluate the mechanical properties of graphene-based composites. The graphene and the matrix are discreted by shell elements and solid elements respectively and the interphase is discreted by isoparmetric graded elements. The RVEs analyzed by isoparmetric graded finite element model (IGFEM) are used to evaluate the mechanical properties. The performance of the proposed model is assessed with some examples available in published literature. The numerical results show that the proposed model has a high accuracy and computational efficiency. It is also demonstrated that the factors, such as interphase material properties and thickness, bonding interface and matrix material properties, can affect the mechanical properties.

Investigation of multibody receding frictional indentation problems of unbonded elastic functionally graded layers

In multibody indentation problems of functionally graded materials, characteristics and gradation of material constituents are greatly influence the response of these problems. The objective of the present study is two folds. The first is to develop a comprehensive and efficient 2D nonlinear inequality constrained variational finite element model capable of analyzing multibody frictional indentation problems of FGMs considering a realistic friction model. The second fold is to investigate the effects of material properties of the constituents of functional graded materials and their gradation, coefficient of friction, and indenter size on the response of multibody frictional indentation systems. For multibody contact configuration, different types of contact at different contact interfaces are modeled in the framework of contact mechanics. To overcome the drawbacks of classical Coulomb's law of friction, friction at the indentation interfaces is modelled using the nonclassical nonlinear local friction model to realize the tangential frictional stress and relative tangential displacements. All frictional contact constraints at all contact interfaces are exactly fulfilled without any penalty parameters. The isoparametric graded finite element formulation is utilized to realize the material properties of FGMs. Application of the developed model is verified with analytical models available in the literature and very well agreement is found. A case study of an unboned elastically graded layered homogeneous elastic substrate indented by a flat rigid indenter subjected to simultaneous normal and tangential loadings is comprehensively investigated. A detailed parametric study is conducted to explore the effect of the layer gradation index, coefficient of friction, and indenter width on the indentation response, which including contact pressure, tangential frictional stress and displacement field at the indenter-graded layer interface and graded layer-elastic substrate interface. Results show that the receding frictional contact response of FGMs can be controlled by appropriate choice of the layer gradient index.

Nonlinear material and geometric analysis of thick functionally graded plates with nonlinear strain hardening using nonlinear finite element method

The present paper deals with three dimensional nonlinear geometrically and elastoplastic analyses of thick functionally graded plates with nonlinear strain hardening. The boundary conditions are assumed to be general. The plate consists of ceramic (SiC) and metal (Al) phases varying smoothly through the thickness. The effective elastic material properties are obtained by Mori-Tanaka scheme where the plastic behavior of the FG plate is described by the von-Mises yield criterion, nonlinear isotropic strain hardening and the Prandtl-Reuss constitutive equations in combination with the TTO model. The governing equations are obtained from the incremental theory of plasticity in conjunction with the Green-Lagrange nonlinear kinematics. In order to investigate the nonlinear elastoplastic response of the FGM, the nonlinear graded finite element method is developed from three-dimensional continuum concepts that admit arbitrarily large displacements and rotations. This formulation has been implemented and the effect of material gradient index, boundary conditions and thickness-to-length ratio on the nonlinear elastoplastic analysis of the FG plate as well as the development of plastic zone are presented and discussed.

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The present paper deals with three dimensional nonlinear geometrically and elastoplastic analyses of thick functionally graded plates with nonlinear strain hardening. The boundary conditions are assumed to be general. The plate consists of ceramic (SiC) and metal (Al) phases varying smoothly through the thickness. The effective elastic material properties are obtained by Mori-Tanaka scheme where the plastic behavior of the FG plate is described by the von-Mises yield criterion, nonlinear isotropic strain hardening and the Prandtl-Reuss constitutive equations in combination with the TTO model. The governing equations are obtained from the incremental theory of plasticity in conjunction with the Green-Lagrange nonlinear kinematics. In order to investigate the nonlinear elastoplastic response of the FGM, the nonlinear graded finite element method is developed from three-dimensional continuum concepts that admit arbitrarily large displacements and rotations. This formulation has been implemented and the effect of material gradient index, boundary conditions and thickness-to-length ratio on the nonlinear elastoplastic analysis of the FG plate as well as the development of plastic zone are presented and discussed.

Modeling and analysis of the nonlinear indentation problems of functionally graded elastic layered solids

In tribological and thermo-electro-mechanical applications with an appropriate gradation of properties, functionally graded materials provide superior performance for damage and wear resistance of contact systems and thermal and mechanical responses for other mechanical devices. This paper presents a nonlinear mixed variational-based computational model for the analysis of nonlinear plane indentation problems of elastic functionally graded layered elastic solids. The functionally graded elastic layer is perfectly bonded to the elastic substrate and is normally loaded by a cylindrical rigid punch. In addition to nonlinearity inherent to the contact problem, the developed model accounts for the geometric nonlinearity in the sense of larger displacements and rotations, but smaller strains. The modulus of elasticity as well as Poisson's ratio of the graded layer are varying along the thickness of the layer according to both exponential and power laws. On the contrary, in the conventional homogeneous finite element formulation the isoparametric graded finite element formulation is adopted on the level of Gaussian integration points to realize the gradation in material properties. The friction at the contact interface is modeled using Coulomb's friction law. Moreover, the inequality contact constraints are exactly satisfied throughout the contact interface by employing the Lagrange multiplier method, where the indentation force and the displacement fields are treated as independent variables. The obtained results of contact pressure, tangential contact stress, stress distribution, and indentation force show a significant influence of the material distribution, gradation law, gradient index, and thickness of the functionally graded layer.

Accurate and efficient thermal stress analyses of functionally graded solids using incompatible graded finite elements

Functionally graded materials have found a wide usage in high temperature applications. The smooth transition from one material to another, in graded materials, may reduce thermal stresses, residual stresses and stress concentration factors as well as utilize properties of both materials. To perform accurate and efficient finite element analysis for heat transfer and transient thermal stress analyses in two-dimensional functionally graded materials, incompatible graded finite elements are developed and verified. User-defined subroutines in ABAQUS are developed to address the gradation of material properties within an element. An emphasis is made on an incompatible six-node graded finite element (QM6) which is accurate and efficient compared to linear four-node (Q4) and quadratic eight-node (Q8) elements. With the help of posteriori error estimation, a critical comparison is made among three types of solid elements. Modified 6-node (QM6) incompatible graded elements provide better accuracy than Q4 elements and take less computational time than Q8 elements, thereby showing QM6 as an optimal element for engineering analysis.

Incompatible Graded Finite Elements for Analysis of Nonhomogeneous Materials

A six-node incompatible graded finite element is developed and studied. Such element is recommended for use since it is more accurate than four-node compatible element and more efficient than eight-node compatible element in two-dimensional plane elasticity. This paper presents comparison between six-node incompatible (QM6) and four-node compatible (Q4) graded elements. Numerical solution is obtained from abaqus using UMAT capability of the software and exact solution is provided as reference for comparison. A graded plate with exponential and linear gradation subjected to traction and bending load is considered. Additionally, three-node triangular (T3) and six-node triangular (T6) graded elements are compared to QM6 element. Incompatible graded element is shown to give better performance in terms of accuracy and computation time over other element formulations for functionally graded materials (FGMs).

On the reduction of stress concentration factor in an infinite panel using different radial functionally graded materials

This work deals with a parametric study of the stress concentration factor in different functionally graded material panels having a central circular hole subjected to a uniaxial tensile, biaxial tensile and shear load. The extended finite element method along with isoparametric graded finite element material properties modelling scheme is used to evaluate the stress concentration factor. Three candidate material gradation models are tested in this parametric study and various metal ceramic functionally graded materials are also tested for stress concentration. The stress concentration factor evaluated numerically for a wide range of Young's modulus ratio and power law index are presented. It has been concluded that the low stress concentration factor can be achieved by choosing the proper material gradation model and their power law index for a fixed value of Young's modulus ratio.

Improved graded finite elements with applications in simulating non-homogeneous elastic, piezoelectric and magneto-electro-elastic materials

Graded finite elements (GFEs) provide a promising way for simulating functionally graded materials. Nevertheless, the existing GFE method takes the conventional isoparametric transform functions as a unified representation for the material non-homogeneity in each element regardless of the practical global distribution forms of material properties. This inevitably leads to a certain difference between the local formulation of material property variation in the elements and the corresponding global gradation patterns. In order to eliminate this difference, an improved GFE algorithm is proposed in the present article. The property distribution in the element is formulated by substituting the isoparametric transform of the coordinates directly into the corresponding global gradation functions. Therefore, the local property distribution is always consistent with the global one. Both the improved six-node triangular elements (T6) and eight-node quadrilateral elements (Q8) are developed for non-homogeneous elastic, piezoelectric and magneto-electro-elastic materials, respectively. Exact solutions in three special cases are presented to make comparison with the numerical results, and the accuracy of the proposed algorithm is verified. It is demonstrated that the improved GFE is an effective method for the numerical simulation of functionally graded materials.

A new graded singular finite element for crack problems in functionally graded materials

A new graded singular finite element is proposed for analyzing crack problems in linear elastic isotropic functionally graded materials (FGMs) with spatially varying elastic parameters. The general formulation of the suggested singular element is obtained by analyzing the crack tip stress field using the Westergaard stress function method. The general shape function is generated by integrating the strains obtained from the stress function. The stiffness matrix for the singular element is then determined using the principle of minimum potential energy. Using the displacement continuity between the singular and the adjacent regular elements, stiffness matrices are assembled. This new element is characterized by containing both singular and higher order terms, which provides more precise description of the crack tip fields. The validity of the new element is demonstrated by comparing with existing solutions. This element is implemented for simulating crack problems in FGMs whose elastic properties vary normal to the crack line. Stress intensity factors, energy release rates, levels of non-singular stresses, and stress distributions near the crack tip are investigated. Numerical results reveal that the introduced graded singular element is more efficient than conventional finite elements as it provides more accurate description of the crack tip field with less efforts.

Modelling functionally graded materials in heat transfer and thermal stress analysis by means of graded finite elements

For better understanding of the behaviour of functionally graded materials (FGM) in high temperature environment, a reliable and efficient numerical tool is required for predictions of heat transfer behaviour and thermally-induced stresses in them. This study presents a finite element formulation of a coupled thermo-mechanical problem in functionally graded metal/ceramic plates. The theoretical framework considers the finite element method (FEM) which is applied to the development of a functionally graded two-dimensional plane strain finite element. The plane strain graded finite element is incorporated within the ABAQUStm code via the combination of user-defined subroutines. The subroutines enable us to program graded mechanical and thermal properties of the FGM as continuous position-dependent functions and, then, to sample them directly at the Gauss integration points of the element. The performance of the developed graded finite element is verified by comparisons with results known in the literature and with calculated using conventional homogeneous elements in a layered model. The solutions of thermo-mechanical problems of functionally graded plates referring to pure mechanical and thermal tasks, and uncoupled and coupled analyses of thermoelasticity are carried out and discussed in the paper.

Analysis of Dynamic Fracture Parameters in Functionally Graded Material Plates with Cracks by Graded Finite Element Method and Virtual Crack Closure Technique

Based on the finite element software ABAQUS and graded element method, we developed a dummy node fracture element, wrote the user subroutines UMAT and UEL, and solved the energy release rate component of functionally graded material (FGM) plates with cracks. An interface element tailored for the virtual crack closure technique (VCCT) was applied. Fixed cracks and moving cracks under dynamic loads were simulated. The results were compared to other VCCT-based analyses. With the implementation of a crack speed function within the element, it can be easily expanded to the cases of varying crack velocities, without convergence difficulty for all cases. Neither singular element nor collapsed element was required. Therefore, due to its simplicity, the VCCT interface element is a potential tool for engineers to conduct dynamic fracture analysis in conjunction with commercial finite element analysis codes.

Three-Dimensional Stress and Free Vibration Analyzes of Functionally Graded Plates with a Circular Hole by the Use of the Graded Finite Element Method

Three-Dimensional Stress and Free Vibration Analyzes of Functionally Graded Plates with a Circular Hole by the Use of the Graded Finite Element Method

Material parameter identification in functionally graded structures using isoparametric graded finite element model

Abstract An identification algorithm based on an isoparametric graded finite element model is developed to identify the material parameters of the plane structure of functionally graded materials (FGMs). The material parameter identification problem is formulated as the problem of minimizing the objective function, which is defined as a square sum of differences between measured displacement and calculated displacement by the isoparametric graded finite element approach. The minimization problem is solved by using the Levenberg-Marquardt method, in which the sensitivity calculation is based on the differentiation of the governing equations of the isoparametric graded finite element model. The validity of this algorithm is illustrated by some numerical experiments. The numerical results reveal that the proposed algorithm not only has high accuracy and stable convergence, but is also robust to the effects of measured displacement noise.

Three dimensional Free Vibration and Transient Analysis of Two Directional Functionally Graded Thick Cylindrical Panels Under Impact Loading

In this paper three dimensional free vibration and transient response of a cylindrical panel made of two directional functionally graded materials (2D-FGMs) based on three dimensional equations of elasticity and subjected to internal impact loading is considered. Material properties vary through both radial and axial directions continuously. The 3D graded finite element method (GFEM) based on Rayleigh-Ritz energy formulation and Newmark direct integration method has been applied to solve the equations in space and time domains. The fundamental normalized natural frequency, time history of displacements and stresses in three directions and velocity of radial stress wave propagation for various values of span angel of cylindrical panel and different power law exponents have been investigated. The present results show that using 2D-FGMs leads to a more flexible design than conventional 1D-FGMs. The GFEM solution have been compared with the results of an FG thick hollow cylinder and an FG curved panel, where a good agreement between them is observed.

Three dimensional graded finite element elasticity shear buckling analysis of FGM annular sector plates

In the present paper, shear buckling analysis of functionally graded annular sector plates is investigated for the first time. A three-dimensional elasticity approach is employed instead of the approximate plate theories. In-plane shear loads have been applied to radial, circumferential, or all edges of annular sector plates. Moreover, buckling of annular sector plates subjected to in-plane shear load at upper/lower surfaces is investigated for the first time. Three different boundary conditions have been examined: (1) Movable simply supported edges. (2) Immovable simply supported radial edges and free circumferential edges. (3) Immovable simply supported circumferential edges and free radial edges. The material properties are assumed to have transverse heterogeneity according to a power law distribution while Poisson's ratio is assumed to be constant. Results are extracted based on the principle of minimum total potential energy and a graded finite element method. Buckling loads are obtained based on a generalized geometric stiffness concept. In addition to different loading and boundary conditions, the effects of material gradient exponent, sector angles, aspect ratio and thickness ratio on the shear buckling loads and mode shapes of FGM annular sector plates have been investigated in detail.