A new beam finite element for the analysis of FGMs

8/3/2019 A new beam finite element for the analysis of FGMs

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Available online at www.sciencedirect.com

International Journal of Mechanical Sciences 45 (2003) 519539

A new beam nite element for the analysis of functionallygraded materials

A. Chakrabortya, S. Gopalakrishnana ;, J.N. Reddyb

aDepartment of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, Indiab

Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA

Received 17 April 2002; received in revised form 12 March 2003; accepted 24 March 2003

Abstract

A new beam element is developed to study the thermoelastic behavior of functionally graded beam structures.

The element is based on the rst-order shear deformation theory and it accounts for varying elastic and

thermal properties along its thickness. The exact solution of static part of the governing dierential equations

is used to construct interpolating polynomials for the element formulation. Consequently, the stiness matrix

has super-convergent property and the element is free of shear locking. Both exponential and power-law

variations of material property distribution are used to examine dierent stress variations. Static, free vibrationand wave propagation problems are considered to highlight the behavioral dierence of functionally graded

material beam with pure metal or pure ceramic beams.

? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Functionally graded materials; Stress pattern; Wave propagation; High frequency; Finite-element method

1. Introduction

An ideal material combines the best properties of metals and ceramicsthe toughness, electricalconductivity, and machinability of metals, and the low density, high strength, high stiness, and

temperature resistance of ceramics. Demands for such materials come from the automotive indus-

try (lightweight and strong materials would increase fuel eciency and last longer), electronics,

telecommunications, and the aerospace and defense industries. In recent years, these types of ad-

vanced materials are no longer dreams but properly conceived and developed. By varying percentage

Corresponding author. Tel.: +91-80-309-2757; fax: +91-80-360-0134.

E-mail address: [email protected] (S. Gopalakrishnan).

0020-7403/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0020-7403(03)00058-4
mailto:[email protected]:[email protected]


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520 A. Chakraborty et al. / International Journal of Mechanical Sciences 45 (2003) 519 539

Nomenclature

u0

axial displacement of reference planew0 transverse displacement of reference plane

rotation about Y-axis

(z) coecient of thermal expansion

T temperature rise/fall

xx axial strain

xz shear strain

E(z) Youngs modulus

G(z) shear modulus

xx axial stress

xz shear stress

P(z) representative of material property parameterPt P(z) at topmost layer

Pb P(z) at bottommost layer

parameter of exponential variation

n parameter of power-law variation

h beam depth

L length of the beam

A cross-sectional area of the beam

S element strain energy functional

%(z) density

T element kinetic energy functionalIk; k = 0; 1; 2 mass moments

Aij ; Bij ; Dij integrated stiness coecients

ATij; BTij integrated elasto-thermal coecients

Nx, Vx and Mx axial force, shear force and bending moment

, shear-exure and axial-exure coupling parameter

non-dimensional parameter in shape function

[(x)] shape function matrix{u} element displacement vector{u} nodal displacement vector

{F

}nodal mechanical force vector

{R} nodal thermal force vector[K]; [M] element stiness and mass matrix

{u} coecient of displacement vector in transformed domainkj wave numbers

cg group wave speed

content of two or more materials spatially, new materials can be formed which will have desired

property gradation in spatial directions. The gradation in properties of the material reduces thermal

stresses, residual stresses, and stress concentration factors.


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A. Chakraborty et al. / International Journal of Mechanical Sciences 45 (2003) 519 539 521

Graded materials are also required to adhere two dierent materials in structures subjected to

dierent loading environments (thermal and mechanical). In the absence of any graded material at

the interface, there is a high chance that debonding will occur at some extreme loading conditions, be

it static, dynamic, or thermal load. Cracks are likely to initiate at interfaces and grow into the weakermaterial section. Residual stresses can be developed due to the dierence in coecients of thermal

expansion of the materials. Gradually varying the volume fraction of the constituents rather than

abruptly changing them over an interface can resolve these problems. Functionally graded materials

(FGMs) are materials or structures in which the material properties vary with location in such a way

as to optimize some function of the overall FGM. The matrix alloy (the metal), the reinforcement

material (the ceramic), the volume, shape, and location of the reinforcement, and the fabrication

method can all be tailored to achieve particular desired properties.

FGM has gained widespread applicability as thermal-barrier structures, wear- and corrosion-resistant

coatings other than joining dissimilar materials [1]. FGMs consisting of metallic and ceramic com-

ponents are well-known to improve the properties of thermal-barrier systems, because cracking ordelamination, which are often observed in conventional two-layer systems, are avoided due to the

smooth transition between the properties of the components in FGMs. In another application of

FGM in thin-walled members like plates and shells, which are used in reactor vessels, turbines and

other machine parts are susceptible to failure from buckling, large amplitude deections, or excessive

stresses induced by thermal or combined thermomechanical loading. Functionally gradient coatings

on these structural elements may help reduce the failures.

The literature on the response of such advanced materials to dynamic and impact loadings (se-

vere mechanical environments) are limited in numbers. Analyses of shear deformable plate with

through-thickness material property variation in the presence of the von Karman non-linearity are

carried out by Reddy and Chin [2], Praveen and Reddy [3], Reddy [4] and Reddy and Hsu [5]. To

the best of authors knowledge, no nite-element formulation is available in the literature for FGM beams. In this paper, we take a novel approach of developing an FGM beam nite element by de-

riving the approximation functions from the exact general solution to the static part of the governing

equations. These solutions are then used to construct accurate shape functions which result in exact

stiness matrix and a mass matrix that captures mass distribution more accurately compared to any

other existing beam nite elements. Thus, the element is an ecient tool for modeling structural

systems to study wave propagation phenomena that results due to high frequency and low duration

forcing (impact loading).

Exact stiness matrices were developed earlier for HermanMindlin rod [6], rst-order shear de-

formable composite beam [7], rst- and higher-order shear deformable isotropic beam [8,9] and

beam with Poissons contraction [10]. In this approach, the shape functions are not only a functionof length of the beam but also depend upon cross-sectional and material properties. The degree of

interpolating polynomials for eld variables is dictated by order of the governing dierential equa-

tion, which attributes to the super-convergent property of the elements (i.e., the nodal values of the

static solution are exact).

Analysis of FGM involves consideration of temperature change during mechanical loading, which

also imparts thermal loading of signicant amount because of mismatch in thermal coecients

between metallic and ceramic materials. Praveen and Reddy [3] have already dealt with this problems

for static and transient loading. El-Abbasi and Meguid [11] analyzed the thermoelastic behavior of

functionally graded plates and shells. In the present work, eect of the temperature rise/fall is


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Fig. 1. (a) Coordinate system and degrees of freedom for the nite element; and (b) beam geometry for modulated pulse

load.

considered by augmenting the thermal strain to the mechanical strain, instead of solving the coupled

thermoelastic equations. Both power law and exponential law are taken for the variation of the

material properties through the depth of the beam.

The paper is organized as follows. In Section 2, the details of formulation of exact beam nite

element is given. Section 3 describes static, free vibration and wave propagation studies using thiselement. In static problems, variations of stresses across depth due to both mechanical and thermal

loadings are investigated. In free vibration analysis, eect of FGM on natural frequencies and its

dependence on through-the-thickness property distribution is studied. In the wave propagation studies,

behavior of bi-material beam fused with FGM and subjected to high-frequency loading is studied.

In particular, its mechanical response is compared with ceramic or steel beam. The eect of FGM

content in shear mode and cut-o frequency is investigated.

2. Finite-element formulation

Considering the rst-order shear deformation (or the Timoshenko beam) theory, the axial and the

transverse displacement eld are expressed as

U(x;y;z;t) = u0(x;t) z(x;t); W(x;y;z;t) = w0(x;t); (1)where u0 and w0 are the axial and transverse displacements in the reference plane, respectively (see

Fig. 1(a)), and z is the thickness coordinate measured from the reference plane. Using Eq. (1), the

linear strains can be written as

xx = u0;x z;x (z) T; xz = + w0;x : (2)


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where the last terms in Eqs. (8) and (10) are zero because of uniformity of the temperature. Asso-

ciated force boundary conditions are

A11u0

;x B11;x AT11 T = Nx; (11)A55(w

0;x ) = Vx;B11u0;x + D11;x + BT11 T = Mx; (12)where the stiness coecients are obtained as

[A11 B11 D11] =

A

E(z)[1 z z2] d A; A55 =

A

G(z) dA; (13)

[AT11 BT11] =

A

E(z)(z)[1 z] dA; (14)

and the mass moments are

[I0 I1 I2] =

A

%(z)[1 z z2] dA: (15)

In Eqs. (11) and (12), Nx, Vx and Mx are, respectively, the axial force, shear force and bending

moment acting at the boundary nodes. The interpolation functions for the displacement eld for the

nite-element formulation are obtained by solving a system of ordinary dierential equations (ODEs)

which is the static part of the governing partial dierential equations (PDEs) given by Eqs. (8)(10).

The exact solution has the form

u0 = c1 + c2x + c3x2; (16)

w0 = c4 + c5x + c6x2 + c7x

3; (17)

= c8 + c9x + c10x2: (18)

From Eqs. (16)(18), we see that the order of interpolation of w0 is one order higher than slope .

This is one of the requirements for the element to be free of shear locking (see Refs. [ 7,9]). The

exact solutions for the displacements have a total of 10 constants and only six boundary conditions

(three degrees of freedom at each node of the element) are available. Hence, there are only six

independent constants. The additional four dependent constants can be expressed in terms of six

other independent constants by substituting Eqs. (16)(18) into Eqs. (8)(10). In doing so, we get

c3 = c10B11=A11; c7 = c10=3; c6 = c9=2; c10 = (c8 c5)=2;or

c3 = (c8 c5)=2; c7 = (c8 c5)=6;where

= B11A55=(A11D11 B211); = A11A55=(A11D11 B211): (19)From relations in Eq. (19), constants c3, c6, c7 and c10 can be written in terms of rest of the constants

and the exact solution takes the form

u0 = c1 + c2x +12

(c8 c5)x2;


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w0 = c4 + c5x +12

c9x2 + 1

6(c8 c5)x3;

= c8 + c9x + 12 (c8 c5)x2: (20)In matrix form

{u} = {u0w0}T = [N(x)]{a}; {a} = {c1; c2; c4; c5; c8; c9}; (21)where [N(x)] is the matrix containing functions of x, i.e x, x2 and x3, and it is of size 3 6. Thecolumn vector {a} of independent constants can be expressed in terms of nodal displacements bysubstituting six displacement boundary conditions while evaluating Eq. (21) for x = 0 (node 1) and

x = L (node 2). The relation is

[G]1 =

N(0)N(L)

; {u} = [G]1{a}; {a} = [G]{u}; (22)

where u={u1 w1 1 u2 w2 2}T is the nodal displacement vector for the element. Now the displace-ments at any point in the element can be expressed in terms of nodal displacements by substituting

Eq. (22) into Eq. (21):

{u} = {u0 w0 }T = [N(x)]{a} = [N(x)][G]{u} = [(x)]{u}: (23)[(x)]=[u(x) w(x) (x)]T, where u(x), w(x) and (x) are the exact shape functions for axial,transverse and rotational degrees of freedom, respectively. They are given in Appendix A.

Substituting the displacement eld from Eq. (21), the force resultants in Eqs. (11)(12) can bewritten in terms of the generalized displacements {a} as

{F} = [ G]{a} + {R}; (24)where

{F} = {Nx(0) Vx(0) Mx(0) Nx(L) Vx(L) Mx(L)}: (25)Substituting Eq. (22) into Eq. (24) we get the relation

{F}

= [ G][G]{

u}

+{R}

= [K]{

u}

+{R}

; (26)

where [K] is the element stiness matrix and {R} is element load vector due to change in temperature.Explicit forms of [K] and {R} are given in Appendix A.

Next, the consistent element mass matrix is computed. It can be expressed as a sum of four

submatrices as shown below:

[M] = [Mu] + [Mw] + [M] + [Mu]: (27)

Here [Mu]; [Mw] and [M] represent the contribution of u, w and degree of freedom to the mass

matrix while [Mu] represents the mass matrix arising due to coupling between u and degrees of


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freedom. These in expanded form can be written as

[Mu] = L

0

I0([

u]

T[

u]) dx; [Mw] =

L

0

I0([

w]

T[

w]) dx; (28)

[M] =

L0

I2([]T[]) dx; (29)

[Mu] = L

0

I1([u]T[] + []T[u]) dx: (30)

We see that unlike conventional elements, the approximation functions of this element are dependent

not only on the length of the element but also on its material and cross-sectional properties. The

stiness matrix developed is exact for point loading as it is derived from displacement eld that

exactly satises the homogeneous form of the governing static dierential equation. Whereas, the

mass distribution is approximate because the approximation functions are not the exact solution ofthe governing PDEs (Eqs. (8)(10)), where inertial terms are present. This particular aspect is very

crucial to the quality of the response predicted by the element. This is because, the order of error

introduced by the approximate stiness matrix, as in the case of other element formulation, is one

order higher than that for mass matrix [17]. In our formulation of the mass matrix, rotary inertia and

the eect of geometric and material asymmetry is taken into account, which signicantly increases

the accuracy.

3. Numerical experiments

The present nite element has been subjected to rigorous numerical testing to establish its validityas error-free nite element for static case and most accurate element for free vibration and wave

propagation problems. In static problems, the authors wish to show the accuracy of the present ele-

ment in the analysis of FGM beams in stress smoothening (which appears due to property mismatch)

when more than one type of material is present in the structures.

3.1. Static problems

The formulated element herein is used to model a bi-material beam where the transition is

made smooth by inserting a thin FGM layer. Materials considered are: (1) steel which has fol-

lowing properties: E= 210 GPa, G = 80 GPa, = 14:0 106

C1

and (2) alumina (Al2O3) whose properties are E = 390 GPa, G = 137 GPa, = 6:9 106C1. Using these materials a function-ally graded cantilever beam of 0:5 m length subjected to transverse load at the tip is considered.

The topmost material is steel which has a thickness of 0:0125 m and bottom layer is alumina of

thickness 0:0325 m. In between these layers there is an FGM layer of 0 :005 m. Material proper-

ties are assumed to vary according to the exponential law. The beam has unit width and there

is no rise in temperature (T = 0). An all-FGM beam is also considered to compare the stress

distributions.

First, a unit transverse load is applied at tip and the stress pattern at the xed end with and

without FGM layer is shown in Fig. 2. From the gure, we see that in the absence of FGM layer,


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-1000 -500 0 500 1000 1500

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Axial Stress (N/m2)

depthz(m)

10 15 20 25 30

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Shear Stress (N/m2)

depthz(m)

( without FGM )

( without FGM )( full FGM )

( full FGM )

Fig. 2. Depthwise stress distribution for transverse load: o denotes bimaterial beam without FGM layer.

the stress distributions are discontinuous. Introduction of a small FGM layer smoothens the axial

stress to the tune of about 300 and 10 N=m2 of shear stress over 0:05 m depth which corresponds

to an axial stress gradient of 6000 Pa=m and a shear stress gradient of 200 Pa=m. Also shown in

both the gures is the stress variation for all-FGM beam which is characterized by a smooth curve

throughout the depth. Similarly, a unit axial load is applied at tip and stress distribution is measuredat tip which is shown in Fig. 3. While shear stress is identically zero for this case, FGM layer

smoothens a jump of 20 N=m2 (stress gradient of 400 Pa=m).

Next the eect of thermal loading is studied. The same cantilever beam is taken but depths are

dierent this time. Here thickness of steel, FGM and alumina layers are 0.02, 0.05 and 0:03 m,

respectively. No external load is applied but temperature is raised by 5C (T). The exponent n

is varied from 1 to 5. The stress distribution is shown in Fig. 4 where the smoothening eect of

FGM layer can be clearly seen. For this case the axial stress distribution varies in the same way as

shear distribution and it shows sensitive dependence on the exponent n. The exponential distribution

is concave as in the previous case.


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3 4 5 6 7

x 1010

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Axial Stress (N/m2)

d

epthz(m)

T = 5 C

2 3 4 5

x 105

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Shear Stress (N/m2)

d

epthz(m)

T = 5 C

Fig. 4. Depthwise stress distribution for thermal load (arrow denotes increase in n) Note: FGM layer has z coordinate

|z|6 0:025 and o denotes exponential distribution.

exponent n and the rate is higher for higher modes, i.e eect of n is more pronounced in higher

modes.

3.3. Wave propagation analysis

The characteristics of the wave propagation problem is that the frequency content of the forc-

ing function (such as high-velocity impact or blast loading) is very high. Hence, unlike conven-

tional structural dynamic problems, all the higher-order modes also participate in the response.

At higher frequencies, the wavelengths are smaller requiring also the element size to be smaller

(in the order of the wavelength). Therefore, FE model for wave propagation has a large system

size.

The behavior of shear deformable composite beam, subjected to high-frequency loading, is rigor-

ously analyzed in Ref. [7]. Behavior of metallic or ceramic beam is the same as that for composite


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0 0.2 0.4 0.6 0.8 10.07

0.072

0.074

0.076

0.078

0.08

0.082

0.084

tfgm

/h

1st natu

ralfrequency(kHz)

exponentialn=1.5n=2

n=3

0 0.2 0.4 0.6 0.8 13.3

3.4

3.5

3.6

3.7

3.8

3.9

4

tfgm

/h

10th

naturalfrequency(kHz)

exponentialn=1.5n=2n=3

Fig. 5. Variation of natural frequencies with increase in FGM content.

with 0 ply-angle. But for FGM beam, cross-section asymmetry with respect to the Y-axis induces

axial-exure coupling and its behavior matches with the cross- or angle-ply composite beam. The

aim of the present section is to show this behavior and to capture dispersiveness that is present

due to material asymmetry. First, the FGM beam is studied to highlight the dierence in axial and

transverse velocity pattern with metallic and ceramic beam. Next, dierent propagating modes are

captured and their occurrences are analyzed in the light of spectral analysis and its dierence in

response to metallic (or ceramic) beam is studied.

3.3.1. Cantilever beam under tip impact load

The aim of this study is to capture the dierence in behavior of FGM beam with all-metallic

or all-ceramic beam. A cantilever beam with length L = 1 m, width b = 0:01 m and overall depth

h = 0:05 m is considered. As in the case of static analysis, materials considered are: (1) steel which

has the following properties: E=210 GPa, G =80 GPa, = 14:0106C and (2) alumina (Al2O3)whose properties are E = 390 GPa, G = 137 GPa, = 6:9 106C. Three beams are considered:(a) all-FGM beam, (b) all-steel beam, (c) all-ceramic beam and (d) steelFGMceramic beam with


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0 50 100 150 200 250 300

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Load(N)

0 20 40 60 80 1000

0.5

1

1.5

2

2.5x 10

-5

Frequency (KHz)

FrequencyAmplitude

Fig. 6. Triangular pulse with Fourier transform inset.

depthwise material distribution same as in static case. An impact load with peak amplitude 1:0 N and

of 50 s duration (shown in Fig. 6) is considered. As seen in this gure, it has a very high-frequency

content (nearly 44 kHz). The beam is modeled with 1000 beam elements resulting in a system size

of 3000 6 in banded form. Newmark time integration scheme with time step of 1 s is used.First, the beam is impacted axially at the tip and the axial velocity is measured at the impact site.

The velocity history for the beams (a)(d) is shown in Fig. 7. It can be observed that reection

from the root in steelFGMceramic beam (beam (d)) occurs later than that of all-ceramic beam

(beam (c)) but before all-steel beam (beam (b)). Also, reection for all-FGM beam (beam (a))

comes later than beam (d). Hence, it can be inferred that axial velocity for FGM beam is greater

than steel beam but lower than all-ceramic beam which is due to magnitude of respective Youngsmoduli. Also, it is evident from the gure that, FGM beam exhibits higher dispersiveness (change in

shape of initial waveform with propagation, here manifested through multiple waveforms just after

the boundary reections), that is higher than both metallic and steel beam, which may be attributed

to both mass and stiness coupling.

Next the same cantilever beam is impacted in the transverse direction at the tip. Fig. 8 shows

the comparison of transverse tip velocity histories. For ceramic beam, transverse group velocity

is very high and reection from xed end comes quickly (around 400 s) whereas for steel it

comes later (at around 700 s) and for FGM beam it comes after 525 s, i.e FGM beam has trans-

verse velocity higher than all-steel beam but lower than all-ceramic beam. Also, initial response


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0 100 200 300 400 500 600-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Time (sec)

Axialvelocity(mm/sec)

(a) all FGM(b) Steel(c) Ceramic(d) Fe-FGM-Al

Fig. 7. Comparison of axial tip velocity for axial load.

amplitude of FGM beam is in between ceramic and steel suggesting its average mass and sti-ness properties. Response of all-FGM beam (a) matches closely with its rareed counterpart (beam

(d)) where there is a dierence of around 100 s between the occurrence of their reections from

root.

3.3.2. Innite beam under modulated pulse

In the previous study, the load contains a spectra of frequency varying from 0 to 40 kHz which

is not suitable for study of individual propagating modes. This is because, capturing the trans-

verse as well as the axial responses, which are essentially dispersive in nature, requires a forc-

ing function, that exists only at a single frequency, specially at very high frequencies. That is,

one requires a forcing function that forces the transverse and axial responses to be dispersive tocapture dierent propagating modes. A modulated narrow banded pulse will satisfy this require-

ments. Present element, being shear deformable, is expected to capture shear propagating mode

in addition to axial and bending modes. Shear mode has a unique characteristics in the way

that it appears only when loading frequency exceeds a certain value called the cut-o frequency.

Cut-o frequency depends upon material properties and its expression can be found in Ref. [7].

For loading frequency less than cut-o frequency, this mode is absent and only bending mode

is visible. Since presence of FGM induces material coupling (both B11 and I1 are non-zero), an

axial load results in the bending, hence the shear mode, while the transverse load will also in-

duce an axial mode in addition to the shear mode. For the mono-material beam, such as steel or


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0 100 200 300 400 500 600 700 800 900 1000-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time ( sec)

Transversevelocity(mm/sec)

(a) all FGM(b) Steel(c) Ceramic(d) Fe-FGM-Al

Fig. 8. Comparison of transverse tip velocity for transverse load.

ceramic, there is no axial-exural coupling and the transverse load induces shear and bending modes

only.

Occurrences of these modes can best be analyzed if dispersion relation and spectrum relation

of the corresponding beam is studied, which is more frequent in the domain of spectral analysis.

Spectrum relation is variation of wavenumbers with frequency, where the inverse of the slope of the

curves gives corresponding group speed. Similarly, dispersion relation is variation of group speed

with frequency. Wavenumbers can be thought as frequency in spatial dimensions, i.e., number of

oscillations in a unit wavelength. Each individual mode (corresponding to each individual wavenum-

ber) propagates with dierent speeds (called phase speeds), but the wave packet (resultant of all

the modes) propagates in a dierent speed which is the group speed. In time domain data, it is thegroup speed that determines the time of appearance of individual propagating modes. Both phase

speed and group speed are functions of frequency. When they are equal the system is non-dispersive

(as in rod model) and when they are not the system is dispersive (beam, plate, etc.). As our forcing

function is monochromatic, we can precisely know the values of the dierent group speeds at that

particular loading frequency. Knowledge of group speed variation, rst of all, will help to identify

each individual propagating mode. Secondly, it will help to verify the accuracy of the FE formula-

tions by predicting the time of occurrence of each individual modes. The spectrum and dispersion

relations can be obtained in the following way. First, eld variables are transformed onto the fre-

quency domain using discrete Fourier transform (DFT). The discretized form of the displacement


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0 10 20 30 40 50 60 70 80 90 100-50

0

50

100

150

200

kFe

sk

fgm

sk

Al2O

3

s

Wavenumberkj

(m-1)

kFe

b

kfgm

b

kFe

a

kAl

2O

3

a

kfgm

a

Frequency (kHz)

Fig. 9. Spectrum relation for dierent materials.

eld in terms of structural frequency (!n) is expressed as

[u0; w0; ](x;t) =

Nn=1

[uj ; wj; j]ei(kjx!nt); (31)

where i =1 and kj is called the wavenumber associated with the jth mode of propagation.

Next, Eq. (31) is substituted in governing dierential equations (Eqs. (8)(10)), and for non-trivial

solution of uj; wj and j, an equation can be obtained involving kj, which in the present case will be

a sixth-order polynomial equation, which when solved gives the spectrum relation. From spectrum

relation, dispersion relation can be obtained using the following formula for group velocity ( cg) (see

Ref. [18]):

cg =9!

9k: (32)

In the present case, four types of beam are considered for analysis where one diers from other in

their material properties. The rst one is an all-ceramic beam, second is the all-steel beam, third one

is an all-FGM beam, while the fourth one is a ceramic + FGM + steel beam, where FGM smoothly

blends all the properties of ceramic to that of steel. For the present case, an innite beam is modeled

with a nite length beam of length 8 m and xed at both ends. The beam is impacted at point A (see

Fig. 1( b)) and the response is measured at point B, where the propagating length allowed is 2 m.


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10 20 30 40 50 60 70 80 900

2000

4000

6000

8000

10000

12000

Frequency kHz

GroupSpeed(m/s)

CFe

b

Cfgm

b

CAl

2O

3

b

CAl

2O

3

s

CAl2O3

a

Cfgm

a

CFe

a

CFe

s Cfgm

s

load

Fig. 10. Dispersion relation for dierent materials: applied load is also shown.

For all materials, the beam is of square cross-section of 0:1 m. Eight thousand elements are used to

model the beam, which results in a system size of [239976]. Material properties are taken as beforeand value of exponent (n) in power law is taken as 1.5. Fig. 9 shows the spectrum relation for three

dierent materials where the subscript denotes material type and superscript denotes propagating

modesa for axial, b for bending and s for shear. Real part of the wavenumber is plotted on the

positive side of ordinate and complex part on the negative side. Those modes which have higher

slopes correspond to lower group speed and it can be said that bending speed of steel is lower than

that of mixed FGM beam, which is again lower than an all-ceramic beam, etc. As seen from the

gure, shear wavenumbers are initially in the complex zone denoting evanescent modes and after

crossing their respective cut-o frequencies, they appear on the real side and propagate. Here, itis clear that FGM beam has cut-o frequency, that is in between steel and ceramic beam. Fig. 10

shows dispersion relation for all three materials. Here, as in the previous case, superscript denotes

modes and subscript denotes materials. It is clear from the gure that axial and bending speed of

FGM beam lie between that of steel and ceramic materials, but the shear speed of FGM material

is greater than the ceramic and converges to shear speed of ceramic at higher frequencies. Also the

initial variation of axial speed of FGM accounts to material coupling which induces dispersiveness in

the response. It is seen from the gure that, for all the materials, the cut-o frequency is well below

40 kHz. Hence a modulated pulse of 50 kHz is applied at point A and response is measured at point

B, which is sucient for appearance of shear mode in all three material sets. The loading spectra


18/21


0 100 200 300 400 500 600 700-2

-1

0

1

-1

0

1

2

s ba

Time (sec)

Tr

ansversevelocity(m/s)10-6

Fe-FGM-Al

FGM-full

Fig. 11. Transverse velocity history at B for modulated transverse load applied at A: lled circle denotes occurrences of

dierent modes predicted by dispersion relation.

are superposed in dispersion relation for easy evaluation of the magnitudes of dierent propagating

speeds, which can be used to check the response obtained from FE analysis.

Responses of the all-FGM beam and blended-FGM beam are shown in Fig. 11. Three separate

wave forms can be seen in that gure which denotes three dierent propagating modes. From dis-

persion relation it is clear that the rst one is shear mode, second one axial mode and third one

is bending mode. Here axial mode appears due to the presence of coupling in FGM beam. From

dispersion relation of ceramic + FGM + steel beam, occurrence of dierent modes is estimated. For

example, shear speed of FGM beam at 50 kHz is around 8245:6 m=s and the time taken for this

mode to travel a length of 2 m is around 242:55 s and added to that initial padding of 100 sin the load history sums up to 342:55 s which is exactly the occurrence of shear mode in Fig.

11. This point and similarly other points (calculated from dispersion relation) for other modes are

plotted with lled circle in that gure, which ensures the validity of present element for analysis

of bi-materials with FGM. It can be seen that there is a little dierence between the behavior of

all-FGM beam and blended-FGM beam. As in the previous case, the shear speed is slightly lower,

while the axial mode travels faster.

Next, responses of steel and ceramic beam are compared to that of an all-FGM beam and the

result is shown in Fig. 12. Dispersion relation predicts that shear speed of FGM beam is greater

than ceramic beam and hence it should appear before shear mode of ceramic beam which is clearly


19/21


0 100 200 300 400 500 600 700 800 900-2

-1

0

0.3

0

0.8

0

0.8

1.6

s b

s b

s ba

Time (sec)

Transversevelocity(m/s)10-6

Al2O

3

FGM

Steel

Fig. 12. Comparison of transverse velocity for modulated transverse load: lled circle denotes occurrences of dierent

modes predicted by dispersion relation.

shown in the gure. All other modes appear according to their velocities given by the dispersion

relation (Fig. 10). Those points of occurrence predicted by dispersion relation are also marked with

lled circles. It is to be noted that ceramic and alumina have no axial modes since no axial-exural

coupling is present for those cases. For them, the rst wave packet is from shear mode and second

one is from bending.

4. Conclusions

In this work, an exact shear deformable nite element for the analysis of FGM is developed.

The element is based on the rst-order shear deformation theory. The consistent mass matrix is

derived using exact approximation based on the exact static deections. Rotary inertia and non-linear

contribution are taken into account. The element is used to study static, free vibration and wave

propagation problems in bi-material beams fused with FGM layer. The smoothening of stresses by

FGM layer is shown in detail for static problem. Eect of the presence of FGM layer in natural

frequencies and in group velocities and wavenumbers is studied.

It has been found that, presence of FGM layer in structures results in signicant dierence in

its response from its parent material beams (steel and ceramic for example) due to the presence of


20/21


coupled stiness and inertial parameters. Static models show that it is an eective way to smoothen

stress jumps in bi-material beams. Its free vibration behavior is remarkable and dierent designs

of these materials will result in signicant variation in its natural frequencies. Its behavior in wave

propagation, in general, is average of the two constitutive materials that it blends. Cut-o frequencyof beam with FGM layer lies between its parent material beams, although shear speed of FGM

applied beam is higher than any of those mono-material beam because of high coupling inertial

terms.

Appendix A.

= B11A55=(A11D11 B211); = A11A55=(A11D11 B211); = 1=(1 + L2=12):The elements of the exact shape function are:

u11 = (1 x=L); u12 = 6(x=L 1)x; u13 = 3(x L)x;

u14 = x=L; u15 = u12; u16 = u13;

w11 = 0; w12 = (L3 + 12L 12x + 2x3 3x2L)=L;

w13 = x(L3 + 6L 6x + x2L 2xL2)=L; w14 = 0;

w15 = x(12 + 2x2 3xL)=L; w16 = x(6L + x2L xL2 + 6x)=L;

11 = 0;

12 = 6x(L + x)=L;13 = (3x2L + L3 + 12L 4xL2 12x)=L; 14 = 0;

15 = 12;16 = x(3xL 2L2 + 12)=L:Non-zero elements of the {R} vector are:

R1 = T AT11; R3 = T BT11; R4 = T AT11; R6 = T BT11:Non-zero entries of the element stiness matrix [K] are:

K11

=A11

L=

K14

; K13

=

B11

L=

K16

;

K22 =A55

L= K25; K23 = A55

2= K26;

K33 =D11

L+

A55 L

4= K66; K34 = K13; K35 = K23;

K36 = D11L

+A55 L

4;

K44 = K11; K46 = K13; K55 = K22; K56 = k23:


21/21


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A new beam finite element for the analysis of FGMs

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