See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/322828386

Geometrically Nonlinear bending analysis of functionally graded beam with

variable thickness by a meshless method

Article  in  Composite Structures · April 2018

DOI: 10.1016/j.compstruct.2018.01.063

CITATIONS

7READS

217

6 authors, including:

Some of the authors of this publication are also working on these related projects:

Composite material View project

Laser Welding Simulation View project

Jun LIN

Shandong University

36 PUBLICATIONS   159 CITATIONS   

SEE PROFILE

Guoqun Zhao

Shandong University

402 PUBLICATIONS   4,506 CITATIONS   

SEE PROFILE

Hakim Naceur

Université Polytechnique Hauts-de-France

147 PUBLICATIONS   1,481 CITATIONS   

SEE PROFILE

Coutellier Daniel

Université Polytechnique Hauts-de-France

182 PUBLICATIONS   666 CITATIONS   

SEE PROFILE

All content following this page was uploaded by Jun LIN on 05 June 2018.

The user has requested enhancement of the downloaded file.

Contents lists available at ScienceDirect

Composite Structures

journal homepage: www.elsevier.com/locate/compstruct

Geometrically nonlinear bending analysis of functionally graded beam withvariable thickness by a meshless method

Jun Lina,b,d, Jiao Lia, Yanjin Guanb,a,⁎, Guoqun Zhaoa, Hakim Naceurc, Daniel Coutellierc

a Key Laboratory for Liquid-Solid Structural Evolution & Processing of Materials (Ministry of Education), Shandong University, Jinan 250061, Chinab Suzhou Institute of Shandong University, Suzhou 215123, Chinac Laboratory LAMIH UMR 8201, University of Valenciennes, Valenciennes 59313, Franced State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan 430074, China

A R T I C L E I N F O

Keywords:Functionally graded beamsSmoothed particle hydrodynamicsGeometrical nonlinearityBending

A B S T R A C T

Geometrically nonlinear bending deformation of Functionally Graded Beams (FGBs) with variable thickness is si-mulated by a meshless Smoothed Hydrodynamic Particle (SPH) method. The material properties of FGB is assumed tobe varied smoothly in the thickness according to exponent-law distribution. To prevent the mesh-distortion in element-based numerical method, meshless SPH method is adopted, where corrective smoothed particle method and total-Lagrangian formulation are employed to improve its precision and stability. To validate the present SPH method,several numerical examples are performed and compared to analytical and finite element solutions.

1. Introduction

Since introduced in thermal barrier on the space shuttle by Japanesescientists in 1984, Functionally Graded Materials (FGMs) have drawn agreat attention of scientists and engineers with wide application inaerospace, automotive, civil, electronics, military, etc. FGMs possessesnoticeable advantage in the smooth gradation of material propertieswhich can reduce the residual stresses and stress concentrations. Thismakes FGMs different from traditional laminated composites in whichthe material properties is abruptly changed across the physical layerinterface leading to large interlaminar stresses and delamination defect.FGMs can be fabricated by desired constituents with designed volumefractions to precisely control its' properties for engineering applications.

With the increasing structural application of FGMs, numerous stu-dies to predict the mechanical response have been performed throughanalytical and numerical methods. The thermo-mechanical behavior ofa functionally graded extensible Timoshenko beam is studied by em-ploying the simplified and original boundary conditions and solving thenonlinear and the linear equilibrium paths [1]. The natural frequenciesof circular and annular thick FG plates composed of two piezoelectriclayers is firstly studied by 3-D Ritz method based on the linear, smallstrain, and 3-D elasticity theory [2]. As the first elasticity solution for atwo-dimensional FGM under applied loading, FG strips and beamssubjected to simple tension or bending moment is theoretically ad-dressed on basis of Airy stress function approach [3]. As for advancedcurved two-directional FGBs with graded material properties along the

axis and thickness directions simultaneously, an analytical model basedon the Euler–Bernoulli theory and classical hairbrush governing equa-tions, is proposed for the flexure prediction [4]. However, the aboveanalytical method is limited for linear solution but difficult to solve thegeometrically nonlinear behaviors of FGMs, which necessitates nu-merical methods like finite element method. Considering warping ef-fects, a continuum mechanics based beam element is formulated toanalyze the geometrically nonlinear responses of 3D FGB under purebending, pure torsion, lateral buckling load and coupled stretching-bending-twisting load [5]. The gradual variation of Poisson’s ratio israrely considered but can be particularly important for correctly re-producing the strain and stress fields along the beam thickness, re-sulting in an investigation of its influences on the displacements, strainsand stress based on 2D beam element [6].

To avoid mesh-related problems, meshless methods have drawnmuch attention and applied in FGMs. A meshless local natural neigh-bour interpolation (MLNNI) method is adopted to study quasi-static andtransient dynamic responses of 2D viscoelastic FG structures [7]. Thecollocation method based on multi-quadric radial basis functions (RBF)is applied to investigate the static response of FGB incorporating hier-archical beam theories [8]. The optimized volume fraction was ob-tained to maximize the first natural frequency of FGB using a meshlessRBF numerical method [9]. Bending analysis of sandwich FG cylindersis performed by asymptotic meshless method using the differential re-producing kernel interpolation and perturbation method [10], wherethe effective material properties are determined using the Mori–Tanaka

https://doi.org/10.1016/j.compstruct.2018.01.063Received 18 September 2017; Received in revised form 25 December 2017; Accepted 19 January 2018

⁎ Corresponding author at: Key Laboratory for Liquid-Solid Structural Evolution & Processing of Materials (Ministry of Education), Shandong University, Jinan 250061, China.E-mail address: guan_yanjin@sdu.edu.cn (Y. Guan).

Composite Structures 189 (2018) 239–246

Available online 31 January 20180263-8223/ © 2018 Elsevier Ltd. All rights reserved.

T

scheme. Subjected to combined thermal and mechanical loads, thermo-elastoplastic analysis of thick FG plates is realized by employing ameshless local Petrov-Galerkin (MLPG) method [11].

In contrast to FGB with uniform thickness, FGB with variablethickness is attracting the interests of researchers in the last severalyears, due to structural matching, strength dispersion, structural weightlightening and economization. On the basis of two-dimensional elasti-city theory and Fourier sinusoidal series expansions, bending analysis isfirstly implemented for FGB with linearly and quadratically varyingthickness [12]. Sari et al. [13] studied the frequency and mode veeringphenomena of axially FG tapered beam based on Timoshenko theory.The Euler-Bernoulli beam theory is also used to analyze the free vi-bration of FGB with variable cross-section on an elastic foundation andspring supports [14]. The free and forced vibrations of axially Ti-moshenko FGB with variable cross-section on elastic foundation areinvestigated [15] analytically and numerically by FEM.

To the best of the authors’ knowledge, the geometrical nonlinearityhas not been considered for FGB with variable thickness and meshlessmethod has not been employed for this type of beams. In this paper, thecomparative study of geometrical linear and nonlinear bending de-formation of FGB with variable thickness is performed using meshlessSmoothed Particle Hydrodynamics (SPH) method. FGM with exponentlaw distribution is considered and two-dimensional elasticity theory isadopted to construct the stress-strain relationship. The Lagrangian SPHmethod [16–18] without using mesh connectivity and background cellis attributed to approximate the strain field. The governing equilibriumequation is also discretized and solved by the proposed SPH methodand then the displacements are updated using central differencemethod.

The rest of this paper is outlined as follows: Section 2 details thematerial properties and governing equations of functionally gradedbeam with variable thickness. The numerical procedure in SPH methodis presented in Section 3. In Section 4, the numerical results are pro-vided and discussed as well as compared to analytical solution in lit-erature and FEM results. Finally the conclusion is drawn in Section 5.

2. Statement of the problem

2.1. Homogenization of the material properties

Considering a Functionally Graded Beam (FGB) with variablethickness shown in Fig. 1, the upper side is horizontal and bears ex-ternal force. The bottom side is smoothly curved and stress free. Theboundary condition taken into account in this paper is simply supportedat two ends of the lower side.

The FGB is often constituted by a mixture of metals and ceramics ofwhich the volume fractions are varied continuously through thethickness direction. The most common method to characterize thegraded properties in this inhomogeneous material is the continuummodel described by the exponent-law or power-law distribution of thecomponents volume fraction. In the present investigation, the materialproperties of FGB is assumed isotropic at any point and varied smoothlyin the thickness according to exponent-law distribution,

=E z E λz H( ) exp( / )0 (1)

=ρ z ρ λz H( ) exp( / )0 (2)

=ν z ν( ) (3)

where E, ν and ρ are Young's modulus, Poisson's ratio and mass density,respectively. The subscript '0′ means the value at the top surface(z= 0). The parameter λ is gradient index to characterize the materialenrichment extent along the thickness direction. The beam is full me-tallic when λ=0, and conversely full ceramic when λ=∞.

2.2. Governing equations

Without loss of generality, the mechanical formulations for general2D linear elastic solid is provided for solving the deformation of thisisotropic inhomogeneous FGM. The elastodynamic equilibrium equa-tion over a domain Ω bounded by a surface Γ can be written

+ = + =σ t b t ρ z u t cu t i jx x x x( , ) ( , ) ( ) ¨ ( , ) ̇ ( , ) (in Ω, , 1,2)ij j i i i, (4)

where σij is stress tensor, bi is body force vector. ui is displacementvectors and c is the damping coefficient. It should be mentioned hereinthat a subscript after a comma denotes partial derivative with respectthe coordinate related to this subscript. Time derivatives are expressedby a dot (first derivative) or two dots (second derivative). The con-ventional Einstein's summation rule over repeated indices is appliedthroughout the paper.

The boundary conditions can be expressed by

=u t u tx x( , ) ( , ) (on Γ )i i u (5)

= =t t σ t n t tx x x x( , ) ( , ) ( ) ( , ) (on Γ)i ij j i t (6)

where ui and ti are the prescribed displacement on the essentialboundary Γu and traction on the natural boundary Γt. ti and n j signifythe surface traction vector and the unit vector outward normal to theboundary Γ ( = ∪Γ Γ Γt u).

For an isotropic elastic material, the most general relation betweenthe stress and strain can be stated as

for plane stress assumption:

⎧⎨⎩

⎫⎬⎭

=−

⎧

⎨⎩

⎫

⎬⎭

⎧⎨⎩

⎫⎬⎭

−

σσσ

Eν

νν

εεγ1

1 01 0

0 0

xxyy

xyν

xxyy

xy2 1

2 (7)

for plane strain assumption:

⎧⎨⎩

⎫⎬⎭

=+ −

⎧

⎨⎩

−−

⎫

⎬⎭

⎧⎨⎩

⎫⎬⎭

−

σσσ

Eν ν

ν νν ν

εεγ(1 )(1 2 )

1 01 0

0 0

xxyy

xyν

xxyy

xy1 2

2 (8)

The kinematic relations for the strain field is related to the corre-sponding displacement

= + +ε u u u u12

( )ij i j j i i k j k, , , , (9)

When the displacements are small enough that the third term in theequation can be negligible, it is considered as geometrically linearproblem.

3. Implementation of SPH method

3.1. Brief introduction of SPH method

Smoothed Particle hydrodynamics (SPH) method was invented forsolving problem in astrophysics [16,17,19–21] and then widely used inthe fluid dynamics [22–24]. The truly meshless feature without usingneither mesh nor background cell, and Lagrangian character to tracethe movement history and crack growth path makes it quite attractivein the field of solid mechanics [25–27]. This also promotes the appli-cation of SPH method for simulating the bending deformation of FGB.

The continuous domain is discretized with a finite number of un-connected particle scattered in space and each SPH particle possessesFig. 1. Functionally graded beam with variable thickness.

J. Lin et al. Composite Structures 189 (2018) 239–246

240

mass, position, velocity, temperature, stress, strain and etc. SPH for-mulation is based on the theory of interpolation, where an arbitraryfunction f can be represented by [28–31]

∫= ′ − ′ ′W h dx x x x xf( ) f( ) ( , ) (10)

where − ′W hx x( , ) is called as kernel function or smoothing function andh is smoothing length. A parameter κ dependent on the function form ofW is introduced to define the radius of non-zero region of the smoothingfunction as κh, i.e. compactly supported domain. Therefore, the integralinterpolation of the function can be approximated by summation pre-sentation in two-dimensional case,

∑==

W Ax x xf( ) f( ) ( )ab

b ab b1

Nb

(11)

where Nb is the number of the neighboring particles b located in thesupport domain of the interested particle a. Wab is the abbreviation of

− hx xW( , )a b and A is the area. By substitution of the function f in Eq.(11) by its derivatives and using integration by parts, one can obtain theintegral and then the summation form of the function derivatives as

∑∇ = ∇=

x xAxf( ) f( ) W ( )ab

b ab b1

Nb

(12)

The smoothing functions have to satisfy several mathematical con-straints of positivity, compact support, normalization and Delta func-tion behavior. It can be constructed in many available forms. In thepresent work, the cubic B-spline function [32] is employed since it isone of the most commonly used smoothing functions,

− = ×⎧

⎨⎪

⎩⎪

− + ⩽ ⩽

− ⩽ ⩽

⩾

x x hπ h

R R R

R R

R

W( , ) 157

if 0 1

(2 ) if 1 2

0 if 2

a b 2

23

2 12

3

16

3

(13)

where = rR h| |/ and = −r x xa b is the position vector between twoneighboring particles. The gradient of this cubic kernel function can begiven as

∇ − = ∇ ×⎧

⎨⎪

⎩⎪

− ⩽ ⩽

− − ⩽ ⩽⩾

x x rhπ h

R R R

R RR

W( , ) 157

2 if 0 1

(2 ) if 1 20 if 2

a b 2 2

32

2

12

2

(14)

It can be easily found that κ=2 when using this cubic-spline kernelfunction.

3.2. Discretization of governing equations

In the following, the kinematic equation and the dynamic balanceequation should be discretized using SPH approximation representationwhen applying it in solid mechanics. However, the inherent imperfec-tions like “inconsistency” [33,34] and “tensile instability” [35,36] arenotorious in the classical SPH formation, which make reduce thecomputation accuracy and stability, even take the computation tobreakout. These problems constraint the application of SPH in solid andthus many remedial measures were proposed [37–44].

In the present work, Corrective Smoothed Particle Method (CSPM)proposed by Chen et al. [39] is employed to conserve the completenessof SPH by taking the Taylor series expansion of the approximatedfunction, which can treat any unsteady boundary value problem withthe Dirichlet and/or von Neumann types of boundary conditions. Thestability of SPH is reinforced by using a Total Lagrangian (TL) for-mulation and [44–46] and artificial viscosity [47]. On contrary to theclassical SPH method in updated Lagrangian formulation, TL formula-tion describes the constitutive relation, kinematic and equilibriumequations in the initially undeformed configuration. There is no need toupdate the support domain and adjacent particles from time to time and

enormously reduce the computation time.For a compact arrangement of this article, the deduce procedure is

passed and one can refer to [39,44]. And then the Total LagrangianCSPM formulation is given directly,

∑

∑= =

=

W A

Af

f

Wa

b

N

b ab b

b

N

ab b

10 0

10 0

b

b

0

0

(15)

∑∇ = − ∇−

=

AKf (f f ) Wab

b a ab b1

1

N

0 0 0

b0

(16)

in which ∇0 signifies the gradient operation with respect to initial ma-terial coordinates. = ∑ − ∇= X X AK ( ( ) W )b b a ab b1

N0 0 0

b0 is introduced to as-sure the first-order completeness of SPH approximation of functiondeterminatives according to the Taylor series expansion method.

In the TL formulation SPH, only the Green Lagrangian strain can beimmediately computed as

⎜ ⎟= ⎛⎝

∂∂

+∂∂

+ ∂∂

∂∂

⎞⎠

E uX

uX

uX

uX

12ij

i

j

j

i

i

k

j

k (17)

where the displacement gradient is approximated by,

∑∂∂

= − ∇−

=

uX

K u u W A( )a b

b a b1

1

N

0 0 0

b

ab

0

(18)

The constitutive relation expressed in Eqs. (7) and (8) link theCauchy stress to the Euler-Almansi strain, therefore the Lagrangianstrain should be transformed to the Euler strain tensor by using thedeformation gradient F

= − −ε F EFT 1 (19)

∑= ∂∂

= − ∇−

=

F xX

x xK ( ) W Aaa b

b a1

1

N

0 0 0b

b0

ab(20)

The equilibrium equation also needs to be rewritten in the initialconfiguration

∂∂

=∂∂

ρ z ut

PX

( ) i ij

j0 2 (21)

where = −σP F Fdet( ) T is the Piola-Kirchhoff stress tensor.Therefore the above equation can be discretized by the TL SPH and

the acceleration can be obtained

∑ ⎜ ⎟∂∂

= ⎛

⎝+ − ⎞

⎠∇

=

P Put ρ ρ

AΠ Wi

a b

aij

a

bij

bab b2

1

N

02

02 ij 0 0 0

b0

(22)

This is of a symmetric form rather than CSPM in order to ensure thelocal momentum balance. The term Π is an artificial viscosity to avoidparticles’ penetration [47],

=⎧

⎨⎩

<

⩾

− +ϕ

ϕΠ

0

0 0ij

α c ϕ β ϕ

ρ ij

ij

ij ij ij

ij

Π Π2

(23)

where = = = =− −

− +

+ + +ϕ h c ρ, , ,ij

h v v x x

x x hij

h hij

c cij

ρ ρ( )( )

| | 0.01 2 2 2ij i j i j

i j ij

i j i j i j2 2 .

αΠ and βΠ are coefficients relative to bulk viscosity and von Neumann-Richtmyer viscosity, respectively. They are set to be 1.0 and 2.0 in thispaper.

The algorithm for computing displacements of the functionallygraded beam with variable thickness can be summarized as:

• Discretize the beam with SPH particles and assign each particle withthe corresponding geometrical parameters and material propertiesE, ρ, ν using Eqs. (1–3).

J. Lin et al. Composite Structures 189 (2018) 239–246

241

• Search the supported domain for each particles and calculate thekernel function W and the derivatives ∇W by employing Eqs. (13)and (14).

• Calculate the strain E, ε and the deformation gradient F using Eqs.(17–20) and then the stress tensor σ using Eqs. (7) and (8).

• Compute the artificial viscosity Π and then the acceleration ofparticles using Eqs. (22) and (23).

• Update the displacement field by integrating the acceleration intime using central difference integration scheme.

4. Numerical results and discussion

In the following numerical examples, the bending deformation ofFunctionally Graded Beams (FGBs) with variable thickness is simulatedby the proposed meshless method. The efficiency of SPH method isvalidated by comparing the results to analytical solution and FEM resultfrom Abaqus commercial software. The influence of geometrical non-linearity, gradient index and depth ratio on the static deflection of FGBare studied. The Young's modulus on the top surface and Poisson's ratioare 2.2×105MPa and 0.3 for all examples. For convenience, the do-main is discretized using a uniformly distributed SPH particles with1mm space distance. All the beams investigated here are simply sup-ported at two ends of the bottom surface. The top surface subjected toan uniform load q0 in the vertical direction.

4.1. Wedge-shaped FGB

The first application deals with a wedge-shaped FGB with linearlyvarying bottom surface as shown in Fig. 2. The geometrical dimensionof the beam is L=2000mm, H=10mm. The depth ratios H1/H=1,1.5, 2 are considered respectively.

Firstly, the geometrical linear analysis under a uniform loadq0= 100 N/mm is performed using the proposed SPH method. Thegradient index λ is fixed as 0.5. A particle distribution of space distance1mm is employed to discretize the wedge-shaped beam, resulting in2000, 3000 and 4000 particles for case H1/H=1, 1.5 and 2 respec-tively.

Also a finite element model is established in the commercial soft-ware ABAQUS for comparison purpose. The integral problem domain isassumed to be stacked by several laminas of 1mm thickness and eachlamina possess the material properties at the midline of the layer. Meshis generated for each independent layer using CPE3 triangle elementinstead of quadrilateral element to prevent low-quality mesh at thesloping side, resulting in 4000, 5195 and 6190 elements for H1/H=1,1.5 and 2 respectively.

The initial and deformed configurations of FGB with H1/H=2 inSPH and ABAQUS are presented in Fig. 3, where a good accordance canbe seen.

The transverse deflection at point x= L/2, z=H/4 is selected tomake comparison with analytical solution in [15], and listed in Table 1.

In the above table, one can find that the transverse deflectionscomputed by the present SPH method and ABAQUS commercial soft-ware match well with the analytical solution. The maximum errors forSPH and ABAQUS solution are only 2.35% and 3.18%.

Then the geometrical nonlinearity is considered. The gradient index

λ is kept as 0.5 and the load q0 is valued from 0 to 1000 N/mm. Noanalytical solution has been concluded in literature due to the diffi-culty, therefore the FE analysis of this problem is implemented inABAQUS. The load-displacement relations for the wedge-shaped FGBare displayed in Fig. 4.

From the above figure, the load-deflection response obtained fromSPH and ABAQUS is similar, which verifies the accuracy of the presentSPH method. When q0 < 200 N/mm, the deformation has a linearcorrelation with the applied load. With the load increasing, a significantgeometrical nonlinearity is appeared. One also can observe that thegradient of the beveled edge has an obvious effect on the bending de-flection, which is reduced with growing of the thickness at the rightside. The deformed configurations with three depth ratios are shown inthe following figure, which explains the good agreement betweenmeshless and finite element analysis (Fig. 5.).

Lastly, the aim is to investigate the influence of the gradient indexesλ on the bending response, considering λ=0, 0.5, 1 and 2. The resultsare plotted in Fig. 6.

It can be found in Fig. 6 that the transverse deflection is decreasedwith gradient index increases. The higher index signifies more ceramiccontained in the FGB which leads to a larger stiffness.

4.2. FGB with parabolic concave lower surface

In this case, a bean with parabolic concave lower surface is con-sidered, as shown in Fig. 7. The beam is simply supported at two ver-texes of the curved side. The maximum depth-to-length ratio of thebeam H/L=0.1 and the total length L=2000mm. Three depth ratioH1/H=0.7, 0.8, 0.9 will be taken account.

Geometrical linear response is firstly computed. The upper surface issubjected to a uniform load q0= 100 N/mm. The transverse deflectionon the cross-section x=0.5L obtained by the present SPH method iscompared to the analytical solution [15] and FE result solved byABAQUS, which is given in Fig. 8.

In Fig. 8, it can be seen that the results from SPH agree well with thesolution from literature and FE analysis. The maximum error is only 7%when H1/H=0.9, since herein the SPH particle of 1mm radius may notprecisely represent the curved surface of very small curvature. Thetransverse deflection is almost same along the section. The deformedconfigurations of the FGBs with three depths are displayed in Fig. 9,testifying the applicability of the present SPH modeling for FGB.

The following analysis take consideration of geometrical non-linearity. The maximum load is 1800 N/mm. FE analysis in ABAQUS isalso carried out for purpose goal. The ABAQUS solver is aborted for caseH1/H=0.7 when the load is greater than 1300 N/mm, due to thedistorted elements. However the SPH code can perform this task. Theresult is shown in Fig. 10.

We can clearly observe from Fig. 10 a good consistency between thesolutions from SPH and ABAQUS. Geometrical nonlinearity influencesenormously on the deformation of the FGB. With increasing the depthratio H1/H, the deflection decreases because it means more materialowned to resist the deformation.

The following figures draw the deformed shapes of FGBs with threedepth ratios (loads corresponding to H1/H=0.7, 0.8 and 0.9 are1200 N/mm, 1800 N/mm and 1800 N/mm, respectively), which furthercertifies the predictive ability of the present SPH model. The ears ap-peared at the supported points would interdict the calculation(Fig. 11.).

The effect of the gradient index on the deformation is lastly focused.λ=0, 0.5, 1 and 2 are introduced to compute the elastic modulus re-spectively. The transverse deflection-load curves are plotted in Fig. 12.

From Fig. 12, the deflection becomes small when gradient indexincreases, due to more ceramic contained in the beam to enlarge itsresistance capacity to deformation. Also one can see that the verticaldisplacement grows up to 0.80 times the structural half-length, whichreveals the good predictive ability of the present SPH method for largeFig. 2. Wedge-shaped FGB with linearly varying bottom surface.

J. Lin et al. Composite Structures 189 (2018) 239–246

242

bending analysis of FGB.

5. Conclusion

In this paper, the geometrical nonlinear deformation of FunctionallyGraded Beam (FGB) with variable thickness is firstly investigated. TheYoung’s modulus and density vary exponentially through the thickness.The beam is simply supported at two ends. The bending deflection ofthe beam under a uniform pressure is computed by a meshlessSmoothed Particle Hydrodynamics (SPH) model, which has been ela-borated. The bending response of two examples are solved by thepresent SPH method and compared to the analytical solution and FEresult obtained by ABAQUS. A good accordance among these resultscan be concluded, which validated the precision and suitability of thepresent SPH method for analyzing large deformation of FGB. Thetransverse deflection decreases when the depth ration and the gradientindex increase.

The present study presents an alternative technique to analyze FGBswith variable thickness, based on meshless SPH method which has beensuccessfully implemented. The proposed method is suitable for homo-geneous and functionally graded structures of uniform and non-uniformsections. Based on this method, further study will focus on the de-formation behavior of two-directional or three-directional functionallygraded structures where the material properties are varied in multi-directions. Furthermore, beam or shell theories will be implementedbased on the presented two-dimensional SPH method to analyze func-tionally graded beams and shells.

Fig. 3. Initial and deformed configurations of FGB in SPH and ABAQUS.

Table 1Transverse deflection w [mm] and its error [%]

Model H1/H=1 H1/H=1.5 H1/H=2

w error w error w error

Analytical 89.09 – 44.59 – 26.00 –Present SPH 87.92 1.31 43.71 1.97 25.39 2.35ABAQUS 86.26 3.18 43.64 2.13 25.67 1.27

Fig. 4. Load-displacement curve of wedge-shaped FGB.

Fig. 5. Deformed configurations of wedge-shaped FGB under q0= 1000 N/mm.

J. Lin et al. Composite Structures 189 (2018) 239–246

243

Acknowledgement

This project is supported by National Natural Science Foundation ofChina (Grant No. 51705291), Jiangsu Province Science Foundation forYouths, China (BK20160371), the financial support from State KeyLaboratory of Materials Processing and Die & Mould Technology,Huazhong University of Science and Technology (P2017-003).

Fig. 6. Load-displacement curve of wedge-shaped FGB with different λ.

Fig. 7. FGB with parabolic concave lower surface.

Fig. 8. Transverse deflection on the cross-section x=0.5L.

Fig. 9. Deformed configurations of FGB with parabolic concave surface.

Fig. 10. Load-displacement curve for FGB with parabolic concave surface.

J. Lin et al. Composite Structures 189 (2018) 239–246

244

References

[1] Dehrouyeh-Semnani AM. On boundary conditions for thermally loaded FG beams.Int J Eng Sci 2017;119:109–27.

[2] Hosseini-Hashemi Sh, Azimzadeh-Monfared M, Rokni Damavandi Taher H. A 3-DRitz solution for free vibration of circular/annular functionally graded plates in-tegrated with piezoelectric layers. Int J Eng Sci 2010;48:1971–84.

[3] Chu P, Li XF, Wu JX, Lee KY. Two-dimensional elasticity solution of elastic stripsand beams made of functionally graded materials under tension and bending. ActaMech 2015;226:2235–53.

[4] Pydah A, Sabale A. Static analysis of bi-directional functionally graded curvedbeams. Compos Struct 2017;160:867–76.

[5] Yoon K, Lee PS, Kim DN. Geometrically nonlinear finite element analysis of func-tionally graded3D beams considering warping effects. Compos Struct2015;132:1231–47.

[6] Paulo Pascon J. Finite element analysis of flexible functionally graded beams withvariable Poisson’s ratio. Eng Computations 2016;33(8):2421–47.

[7] Chen SS, Xu CJ, Tong GS. A meshless local natural neighbour interpolation methodto modeling of functionally graded viscoelastic materials. Eng Anal Boundary Elem2015;52:92–8.

[8] Giunta G, Belouettar S, Ferreira AJM. A static analysis of three-dimensional func-tionally graded beams by hierarchical modelling and a collocation meshless solutionmethod. Acta Mech 2016;227:969–91.

[9] Roque CMC, Martins PALS. Differential evolution for optimization of functionallygraded beams. Compos Struct 2015;133:1191–7.

[10] Wu CP, Jiang RY. An asymptotic meshless method for sandwich functionally gradedcircular hollow cylinders with various boundary conditions. J Sandwich StructMater 2015;17(5):469–510.

[11] Vaghefi R, Hematiyan MR, Nayebi A. Three-dimensional thermo-elastoplasticanalysis of thick functionally graded plates using the meshless local Petrov-Galerkinmethod. Eng Anal Boundary Elem 2016;71:34–49.

[12] Sari M, Shaat M, Abdelkefi A. Frequency and mode veering phenomena of axiallyfunctionally graded non-uniform beams with nonlocal residuals. Compos Struct2017;163:280–92.

[13] Duy HT, Van TN, Noh HC. Eigen analysis of functionally graded beams with vari-able cross-section resting on elastic supports and elastic foundation. Struct EngMech 2014;52(5):1033–49.

[14] Calim FF. Transient analysis of axially functionally graded Timoshenko beams withvariable cross-section. Compos B 2016;98:472–83.

[15] Xu Y, Yu T, Zhou D. Two-dimensional elasticity solution for bending of functionallygraded beams with variable thickness. Meccanica 2014;49:2479–89.

[16] Lucy LB. A numerical approach to the testing of the fission hypothesis. Astron J1977;82:1013–24.

[17] Gingold RA, Monaghan JJ. Smoothed Particle Hydrodynamics: theory and appli-cation to non-spherical stars. MNRAS 1977;1181:375–89.

[18] Monaghan JJ. Smoothed particle hydrodynamics and its diverse applications. AnnuRev Fluid Mech 2012;2012(44):323–46.

[19] Gabor JM, Capelo PR, Volonteri M, Bournaud F, Bellovary J, Governato F, et al.Comparison of black hole growth in galaxy mergers with gasoline and Ramses. A &A 2016;592:A62.

[20] Biffi V, Borgani S, Murante G, Rasia E, Planelles S, Granato GL, et al. On the natureof hydrostatic equilibrium in galaxy clusters. Astrophys J 2016;827:112.

[21] Nelson AF, Marzari F. Dynamics of circumstellar disks III: the case of GG Tau A.Astrophys J 2016;827(2):93.

[22] Jonsson P, Andreasson P, Hellström JGI, Jonsén P, Staffan Lundströma T. SmoothedParticle Hydrodynamic simulation of hydraulic jump using periodic open bound-aries. Appl Math Model 2016;40:8391–405.

[23] Didier E, Neves DRCB, Teixeira PRF, Dias J, Neves MG. Smoothed particle hydro-dynamics numerical model for modeling an oscillating water chamber. Ocean Eng2016;123:397–410.

[24] Leble V, Barakos G. Demonstration of a coupled floating offshore wind turbineanalysis with high-fidelity methods. J Fluids Struct 2016;62:272–93.

[25] Dai Z, Huang Y, Cheng H, Xu Q. SPH model for fluid-structure interaction and itsapplication to debris flow impact estimation. Landslides 2017;14:917–28.

[26] Lin J, Guan Y, Zhao G, Naceur H, Lu P. Topology optimization of plane structuresusing Smoothed Particle Hydrodynamics method. Int J Numer Meth Eng2017;110:726–44.

[27] Vignjevic R, Djordjevic N, Gemkow S, De Vuyst T, Campbell J. SPH as a nonlocalregularisation method: solution for instabilities due to strain-softening. ComputMethods Appl Mech Eng 2014;277:281–304.

[28] Liu GR, Liu MB. Smoothed particle hydrodynamics: a meshfree particle method.Singapore: World Scientific; 2003. 978-9812384560.

[29] Li S, Liu WK. Meshfree particle methods. Berlin Heidelberg New York: Springer;2004. ISBN-10: 3540222561.

[30] Ferreira AJM, Kansa EJ, Fasshauer GE. Progress on meshless methods. Netherlands:Springer; 2009. ISBN-10: 9048179971.

[31] Liu MB, Liu GR. Smoothed particle hydrodynamics (SPH): an overview and recentdevelopments. Arch Comput Methods Eng 2010;17(1):25–76.

[32] Monaghan JJ. Particle methods for hydrodynamics. Comput Phys Rep

Fig. 11. Deformed configurations of FGB with parabolic concave surface.

Fig. 12. Load-displacement curve of FGB with parabolic concave surface of different λ.

J. Lin et al. Composite Structures 189 (2018) 239–246

245

1985;1985(3):71–124.[33] Randles PW, Libersky LD. Smoothed particle hydrodynamics: some recent im-

provements and applications. Comput Methods Appl Mech Eng1996;139(1):375–408.

[34] Jun S, Liu WK, Belytschko T. Explicit Reproducing Kernel Particle Methods for largedeformation problems. Int J Numer Meth Eng 1998;41:137–66.

[35] Swegle JW, Hicks DL, Attaway SW. Smoothed particle hydrodynamics stabilityanalysis. J Comput Phys 1995;116(1):123–34.

[36] Balsara DS. Von Neumann stability analysis of smoothed particle hydrodynamics –suggestions for optimal algorithms. J Comput Phys 1995;121:357–72.

[37] Dilts GA. Moving-least-squares-particle hydrodynamics—I. Consistency and stabi-lity. Int J Numer Meth Eng 1999;44:1115–55.

[38] Bonet J, Kulasegaram S. Correction and stabilization of smooth particle hydro-dynamics methods with applications in metal forming simulations. Int J NumerMeth Eng 2000;47(6):1189–214.

[39] Chen JK, Beraun JE, Carney TC. A corrective smoothed particle method forboundary value problems in heat conduction. Int J Numer Meth Eng1999;46:231–52.

[40] Zhang GM, Batra RC. Modified smoothed particle hydrodynamics method and itsapplication to transient problems. Comput Mech 2004;34:137–46.

[41] Batra RC, Zhang GM. Modified Smoothed Particle Hydrodynamics (MSPH) basisfunctions for meshless methods, and their application to axisymmetric Taylor im-pact test. J Comput Phys 2008;227:1962–81.

[42] Adams B, Wicke M. Meshless approximation methods and applications in physicsbased modeling and animation. In: Eurographics 2009 Tutorials, Seiten; 2009. p.213–239.

[43] Monaghan JJ. SPH without a tensile instability. J Comput Phys 2000;159:290–311.[44] Belytschko T, Guo Y, Liu WK, Xiao SP. A unified stability analysis of meshless

particle methods. Int J Numer Meth Eng 2000;48:1359–400.[45] Maurel B, Combescure A. An SPH shell formulation for plasticity and fracture

analysis in explicit dynamic. Int J Numer Meth Eng 2008;76:949–71.[46] Lin J, Naceur H, Laksimi A, Coutellier D. On the implementation of a nonlinear

shell-based SPH method for thin multilayered structures. Compos Struct2014;108:905–14.

[47] Monaghan JJ, Gingold RA. Shock simulation by the particle method of SPH. JComput Phys 1983;52(2):374–81.

J. Lin et al. Composite Structures 189 (2018) 239–246

246

View publication statsView publication stats