Summary of doctoral thesis in materials science: Finite element models in vibration analysis of two-dimensional functionally graded beams

This thesis aims to develop finite element models for studying vibration of the 2D-FGM beam. These models require high reliability, good convergence speed and be able to evaluate the influence of material parameters, geometric parameters as well as being able to simulate the effect of shear deformation on vibration characteristics and dynamic responses of the 2D-FGM beam.

TRAINING SCIENCE AND TECHNOLOGY

GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY

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TRAN THI THOM

FINITE ELEMENT MODELS IN VIBRATION ANALYSIS OF TWO-DIMENSIONAL FUNCTIONALLY GRADED BEAMS

Major: Mechanics of Solid

Supervisors: 1 Assoc Prof Dr Nguyen Dinh Kien

2 Assoc Prof Dr Nguyen Xuan Thanh

Reviewer 1: Prof Dr Hoang Xuan Luong

Reviewer 2: Prof Dr Pham Chi Vinh

Reviewer 3: Assoc Prof Dr Phan Bui Khoi

Thesis is defended at Graduate University Science and Vietnam Academy of Science and Technology at … , on …

Technology-Hardcopy of the thesis be found at :

- Library of Graduate University Science and Technology

- Vietnam national library

1 The necessity of the thesis

Publications on vibration of the beams are most relevant to FGM beamswith material properties varying in one spatial direction only, such as thethickness or longitudinal direction There are practical circumstances,

in which the unidirectional FGMs may not be so appropriate to resistmulti-directional variations of thermal and mechanical loadings Optimiz-ing durability and structural weight by changing the volume fraction ofFGM’s component materials in many different spatial directions is a mat-ter of practical significance, being scientifically recognized by the world’sscientists, especially Japanese researchers in recent years Thus, struc-tural analysis with effective material properties varying in many differentdirections in general and the vibration of FGM beams with effective mate-rial properties varying in both the thickness and longitudinal directions ofbeams (2D-FGM beams) in particular, has scientific significance, derivedfrom the actual needs It should be noted that when the material properties

of the 2D-FGM beam vary in longitudinal direction, the coefficients in thedifferential equation of beam motion are functions of spatial coordinatesalong the beam axis Therefore analytical methods are getting difficult toanalyze vibration of the 2D-FGM beam Finite element method (FEM),with many strengths in structural analysis, is the first choice to replacetraditional analytical methods in studying this problem Developing thefinite element models, that means setting up the stiffness and mass ma-trices, used in the analysis of vibrations of the 2D-FGM beam is a mat-ter of scientific significance, contributing to promoting the application ofFGM materials into practice From the above analysis, author has selected

the topic: Finite element models in vibration analysis of two-dimensional

functionally graded beams as the research topic for this thesis.

2 Thesis objective

This thesis aims to develop finite element models for studying tion of the 2D-FGM beam These models require high reliability, goodconvergence speed and be able to evaluate the influence of material pa-rameters, geometric parameters as well as being able to simulate the effect

vibra-of shear deformation on vibration characteristics and dynamic responses

of the 2D-FGM beam

3 Content of the thesis

Four main research contents are presented in four chapters of the sis Specifically, Chapter 1 presents an overview of domestic and for-eign studies on the 1D and 2D-FGM beam structures Chapter 2 pro-poses mathematical model and mechanical characteristics for the 2D-FGM beam The equations for mathematical modeling are obtained based

the-on two kinds of shear deformatithe-on theories, namely the first shear formation theory and the improved third-order shear deformation theory.Chapter 3 presents the construction of FEM models based on differentbeam theories and interpolation functions Chapter 4 illustrates the nu-merical results obtained from the analysis of specific problems

de-Chapter 1 OVERVIEW

This chapter presents an overview of domestic and foreign regime of searches on the analysis of FGM beams The analytical results are dis-cussed on the basis of two research methods: analytic method and nu-merical method The analysis of the overview shows that the numericalmethod in which FEM method is necessary is to replace traditional ana-lytical methods in analyzing 2D-FGM structure in general and vibration

re-of the 2D-FGM beam in particular Based on the overall evaluation, thethesis has selected the research topic and proposed research issues in de-tails

Chapter 2 GOVERNING EQUATIONS

This chapter presents mathematical model and mechanical tics for the 2D-FGM beam The basic equations of beams are set up based

characteris-on two kinds of shear deformaticharacteris-on theories, namely the first shear mation theory (FSDT) and the improved third-order shear deformationtheory (ITSDT) proposed by Shi [40] In particular, according to ITSDT,basic equations are built based on two representations, using the cross-sectional rotationθ or the transverse shear rotationγ0as an independentfunction The effect of temperature and the change of the cross-sectionare also considered in the equations

defor-2.1 The 2D-FGM beam model

The beam is assumed to be formed from four distinct constituent rials, two ceramics (referred to as ceramic1-C1 and ceramic2-C2) and twometals (referred to as metal1-M1 and metal2-M2) whose volume fraction

mate-varies in both the thickness and longitudinal directions as follows:

VC1= z

h+12

n z

x L

n zh

n z

x L

h b y z

Fig 2.1 The 2D-FGM beam model

In this thesis, the effective material properties P (such as Youngsmodulus, shear modulus, mass density, etc.) for the beam are evaluated

by the Voigt model as:

P= VC1PC1+VC2PC2+VM1PM1+VM2PM2 (2.2)When the beam is in thermal environment, the effective properties ofbeams depend not only on the properties of the component materials butalso on the ambient temperature Then, one can write the expression forthe effective properties of the beam exactly as follows:

n z

+ PM1(T )

h

PC2(T ) − PM2(T )i z

h+12

n z

+ PM2(T )



x L

n x

(2.4)

For some specific cases, such as nx = 0 or n z= 0, or C1 and C2 areidentical, and M1 is the same as M2, the beam model in this thesis re-duces to the 1D-FGM beam model Thus, author can verification theFEM model of the thesis by comparing with the results of the 1D-FGMbeam analysis when there is no numerical result of the 2D-FGM beam Itsimportant to note that the mass density is considered to be temperature-independent [41].

The properties of constituent materials depend on temperature by anonlinear function of environment temperature [125]:

P= P0(P−1T−1+ 1 + P1T+ P2T2+ P3T3) (2.7)This thesis studies the 2D-FGM beam with the width and height arelinear changes in beam axis, means tapered beams, with the followingthree tapered cases [138]:

Timo-2.3 Equations based on FSDT

Obtaining basic equations and energy expressions based on FSDT andITSDT theory is similar, so Section 2.4 presents in more detail the process

of setting up equations based on ITSDT

2.4 Equations based on ITSDT

2.4.1 Expression equations according toθ

From the displacement field, this thesis obtains expressions for strainsand stresses of the beam Then, the conventional elastic strain energy, UB

are mass moments.

The beam rigidities and mass moments of the beam are in the ing forms:

with AC1M1i j , BC1M1

i j are the rigidities of 1D-FGM beam composed of C1

and M1; AC2M2i j , BC2M2

i j are the rigidities of 1D-FGM beam composed of

C2 and M2 Noting that rigidities of 1D-FGM beam are functions of z

only, the explicit expressions for this rigidities can easily be obtained

2.4.2 Expression equations according toγ0

Using a notation for the transverse shear rotation (also known as sic shear rotation),γ0= w0,x+θas an independent function, the axial andtransverse displacements in (2.13) can be rewritten in the following form

2.5 Initial thermal stress

Assuming the beam is free stress at the reference temperature T0 and

it is subjected to thermal stress due to the temperature change The initialthermal stress resulted from a temperature∆T is given by [18, 70]:

σT

xx = −E(x, z, T )α(x, z, T )∆T (2.41)

in which elastic modulus E (x, z, T ) and thermal expansionα(x, z, T ) are

obtained from Eq.(2.4)

The strain energy caused by the initial thermal stressσT

xxhas the form[18, 65]:

UT=12

The total strain energy resulted from conventional elastic strain energy

UB, and strain energy due to initial thermal stress UT[70]

2.6 Potential of external load

The external load considered in the present thesis is a single movingconstant force with uniform velocity The force is assumed to cause bend-ing only for beams The potential of this moving force can be written inthe following form

V = −Pw0(x,t)δhx − s(t)i (2.44)where δ(.) is delta Dirac function; x is the abscissa measured from the left end of the beam to the position of the load P, t is current time calcu- lated from the time when the load P enters the beam, and s (t) = vt is the distance which the load P can travel.

2.7 Equations of motion

In this section, author presents the equations of motion based on ITSDTwith γ0 being the independent function Motion equations for beamsbased on FSDT and ITSDT with θ is independent function that can beobtained in the same way Applying Hamiltons principle, one obtainedthe motion equations system for the 2D-FGM beam placed in the temper-ature environment under a moving force as follows:

Notice that the coefficients in the system of differential equations of

motion are the rigidities and mass moments of the beam, which are the

functions of the spatial variable according to the length of the beam and

the temperature, thus solving this system using analytic method is

diffi-cult FEM was selected in this thesis to investigate the vibration

charac-teristics of beams

Conclusion of Chapter 2

Chapter 2 has established basic equations for the 2D-FGM beam based

on two kinds of shear deformation theories, namely FSDT and ITSDT

The effect of temperature and the change of the cross-section is

consid-ered in establishing the basic equations Energy expressions are presented

in detail for both FSDT and ITSDT in Chapter 2 In particular, with

ITSDT, basic equations and energy expressions are established on the

cross-sectional rotationθ or the transverse shear rotationγ0 as

indepen-dent functions The expression for the strain energy due to the

tempera-ture rise and the potential energy expression of the moving force are also

mentioned in this Chapter Equations of motion for the 2D-FGM beam

are also presented using ITSDT with γ0 as independent function These

energy expressions are used to obtain the stiffness matrices and mass

ma-trices used in the vibration analysis of the 2D-FGM beam in Chapter 3

Chapter 3 FINITE ELEMENT MODELS

This chapter builds finite element (FE) models, means that establish

expressions for stiffness matrices and mass matrices for a characteristic

element of the 2D-FGM beam The FE model is constructed from the

energy expressions received by using the two beam theories in Chapter

2 Different shape functions are selected appropriately so that beam

ele-ments get high reliability and good convergence speed Nodal load vector

and numerical procedure used in vibration analysis of the 2D-FGM beamare mentioned at the end of the chapter.

3.1 Model of finite element beams based on FSDT

This model constructed from Kosmatka polynomials referred as FBKo

in this thesis can be avoided the shear-locking problem In addition, thismodel has a high convergence speed and reliability in calculating the nat-ural frequencies of the beam However, the FBKo model with 6 d.o.f hasthe disadvantage that the Kosmatka polynomials must recalculate eachtime the element mesh changes, thus time-consuming calculations The

FE model uses hierarchical functions, referred as FBHi model in the sis, which is one of the options to overcome the above disadvantages.Recently, hierarchical functions are used to develop the FEM model in1D-FGM beam analysis (such as Bui Van Tuyen’s thesis) Based on theenergy expressions received in Chapter 2, the thesis has built FBKo modeland FBHi model using the Kosmatka function and hierarchical interpola-tion functions, respectively The process of building FE models is similar,Section 3.2 will presents in detail the construction of stiffness and massmatrices for a characteristic element based on ITSDT

the-3.2 Model of finite element beams based on ITSDT

With two representations of the displacement field, two FEM modelscorresponding to these two representations will be constructed below Forconvenience, in the thesis, FEM model uses the cross-sectional rotationθ

as the independent function is called TBSθ model, FEM model uses thetransverse shear rotation as the independent function is called TBSγ

for u0, w0 andθ Herein, linear shape functions are used for the axial

displacement u0(x,t) and the cross-section rotationθ(x,t), Hermite shape functions are employed for the transverse displacement w0(x,t).

With the interpolation scheme, one can write the expression for the formation components in the form of a matrix through a nodal displace-ments vector (3.28) as follows

o

(3.34)The elastic strain energy of the beam UB in Eq.(2.27) can be written

in the form

UB=12

T = 12

nE

in which the element consistent mass matrix is in the form

m = m11uu+ m12uθ+ m22θθ+ m34uγ+ m44θγ+ m66γγ+ m11ww (3.37)with

u0= NudSγ, w

0= NwdSγ γ0= NγdSγ (3.40)

with Nu, Nw and Nγ are the matrices of shape functions for u0, w0 and

γ0, respectively Herein, linear shape functions are used for the axial

displacement u0(x,t) and the transverse shear rotationγ0, Hermite shape

functions are employed for the transverse displacement w0(x,t) The

con-struction of element stiffness and mass matrices are completely similar toTBSθmodel

3.3 Element stiffness matrix due to initial thermal stress

Using the interpolation functions for transverse displacement w0(x,t),

one can write expressions for the strain energy due to the temperature rise(2.42) in the matrix form as follows

UT= 12

The only difference is that the difference of the shape functions Nw is

chosen for w0(x,t) leading to the difference of the strain-displacement

matrix Bt = (N w),xin (3.45)

3.4 Discretized equations of motion

Ignoring damping effect of the beam, the equations of motion for FGM beam can be written in the context of the finite element analysisas

in which D, ¨ D are, respectively, the vectors of structural nodal

displace-ments and accelerations, K, M, Fex are the stiffness matrices due to thebeam deformation and temperature rise, the mass matrix and the nodalload vector of the structure, respectively

In the free vibration analysis, the right-hand side of (3.49) is set to zero

Conclusion of Chapter 3

Chapter 3 builds FE model for a two-node element based on two kinds

of shear deformation theories for beams Based on FSDT, FE models areconstructed by using two different shape functions, such as the Kosmatkafunction and hierarchical shape functions Based on ITSDT, FE modelsare constructed by linear and Hermite shape functions The expression forstiffness and mass matrix for the models based on ITSDT is built on thebasis of considering the cross-section rotation or transverse shear rotation

as independent functions The expression for the stiffness matrix due totemperature rise and the vector of nodal force is also built into Chapter

Chapter 4 NUMERICAL RESULTS AND DISCUSSION

The numerical results are presented on the basis of analyzing threeproblems: (1) Free vibration analysis of the 2D-FGM beam in thermalenvironment; (2) Free vibration analysis of the tapered 2D-FGM beam;(3) Forced vibration analysis of the 2D-FGM beam excited by a movingforce From the numerical results obtained, some conclusions relate tothe influence of the material parameter, the taper ratio, aspect ratio andtemperature rise on the fundamental frequency and the vibration mode to

be extracted Dynamic behaviour of 2D-FDM beams under the action ofmoving force are also discussed in Chapter

4.1 Validation and convergence of FE models

4.1.1 Convergence of FE models