Modelling of Functionally Graded and Layered

Materials With Space-Time Discontinuous Galerkin

Method

H. G. Aksoy∗ and E. Senocak†

Istanbul Technical University, Istanbul, 34437, Turkey

Mechanical behaviour of a materials under impact loads gathers attention by the re-searchers due its military and civil applications. It is required that big amount of energyis dissipated to a large volume in a very short time in materials under impact loads. Thismakes conservation properties of numerical methods used to model the impact problemsimportant for researchers. In the recent years discontinuous Galerkin method gathersattention due to its conservation properties. In this study mechanical behaviour of Func-tionally Graded Materials (FGMs) and layered materials is modelled using space-timediscontinuous Galerkin method. It is seen that space-time discontinuous Galerkin methodis successful in modelling the FGMs and layered materials.

Nomenclature

ρ Densityu Displacementv Velocityw Weight functionσ Cauchy stress tensorb Body forcest Timen Unit normalh Specified displacementg Specified traction forceΓ BoundaryΩ Domainε Strain tensorE Stiffness tensorλ, µ Lame coefficientsE Young’s modulusν Poisson’s ratiof Volume fractiondt time stepdx Element sizeSubscripte Element of interestnb Neighboring elementint InterfaceD DirichletN Neumman

∗Ph. D. candidate, Istanbul Technical University, Istanbul-Turkey. e-mail:aksoyhus@itu.edu.tr†Assoc. Prof., Istanbul Technical University, Department of Mechanical Engineering, Gumussuyu, 34437 Istanbul, Turkey.

e-mail:senocak@itu.edu.tr. Tel: +90 212 293 1300x2722. Fax: +90 212 245 07 95.

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46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference18 - 21 April 2005, Austin, Texas

AIAA 2005-1924

Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction

Two main approaches has been used in designing the composite plates. One is to use layered compositeplates consisting of several layers made of different materials. Second one is to use FGMs.

In modelling the layered media most of the attention is paid to the defining the effective material prop-erties. Many theoretical and experimental studies have been done on this topic.1,2 The reader can consultto the recent review on defining the effective material properties of laminated composite plates by Ref. 3.The use of layered media in designing composite plates has some disadvantages. Sudden change in materialproperties causes stress concentration in the material4 which may cause failure. FGMs have begun to be usedin engineering designs in which the material properties vary gradually.5 Therefore no stress concentrationoccurs due to sudden change of material properties. These properties makes the FGMs attractive for manyapplications. Many researchers has begun to study on modelling the mechanical behaviour of FGMs. FEMis used for modelling the behaviour of FGM under suddenly applied load in Ref. 6. In the study it is assumedthat material properties vary linearly. Different models are compared in determining the effective materialproperties for modelling composites in Ref. 7. They concluded materials with gradually changing materialproperties are more suitable for material problems in which the amplitude of the stress plays the key role.Layered metal-ceramic composites and composites with randomly embedded ceramic particles in a metalmatrix are analyzed in Ref. 8. It is shown that in the FGM the gradual change of material properties givesmore physical results then using average material properties in Ref. 8. Plate impact experiments are madeon aluminum nitride ceramic tiles bonded with thin silicon rubber at velocities ranging from 14-58 m/secin Ref. 9. Results are compared with the analytical solution. They concluded that the pulse is spreadedradially by the flexural waves.

Modelling of both FGMs and layered media are challenging task. Modelling of graded material increasesthe computational cost whilst modelling the layered media degrades the accuracy due to the sudden change ofmaterial properties. Thus degradation of numerical accuracy affects the conservation of energy, momentumand angular momentum. Conservation properties of numerical methods are important for impact problems.Therefore in the recent years more attention paid to the research on the conservative numerical methods fortime integration and discretization in space.

In the field of computational mechanics researchers are focussed on the conservation properties andstability of the time integration algorithms. Newmark time integration method10 is one of the most widelyused time integration method in structural dynamics. On behalf of this, Newmark family of algorithmsare not energy and angular momentum conserving.11 Energy preserving algorithms are developed by theresearchers.12 Thus energy preserving schemes are lack of high frequency dissipation,13 which is necessary fordamping high frequency oscillations in the numerical solution. Space-time finite element method is attractivefor researchers due to the stability properties. On the other hand time finite element methods are not energyconserving.14 One approach in discretizing the balance equations is discontinuous Galerkin method.15 TimeDiscontinuous Galerkin method (TDG) has begun to be used in elastodynamics.16 It’s small phase errorand has small dissipation error at high frequency regime which damps the high frequency oscillations in thenumerical solution17 makes it attractive for wide range of structural problems.

Discontinuous Galerkin method (DGM) is widely used in fluid mechanics problems in discretizing firstorder partial differential equations (PDEs) due to its local and global conservation properties for spatialand temporal discretization.18 As it allows discontinuities at the element interfaces it is advantageous touse DGM for shock wave propagation problems. More recently DGM for the discretization of second orderPDEs has been developed.19,20 Comparison of different approaches used to discretize second order PDEswith DGM can be found in the literature.21,22 Another advantage of the DGM is continuously changingmaterial properties can be defined in an element for FGMs and sudden change of the material properties areallowed at the element interfaces for layered materials.

The aim of this study is to compare the dynamical behaviour of layered composites and composites withgradually changing material properties under impact loads for one and two dimensional problems. We usespace-time discontinuous Galerkin formulation which is given in Ref. 23 for modelling the problems. Onedimensional numerical results are compared with the analytical results of Ref. 24. Two dimensional axi-symmetric plate is also studied which is suddenly loaded from its center. In section II on the following pagegoverning equations for linear elastodynamics are presented. In section III on the next page formulationfor space-time discontinuous Galerkin method is presented. Details of numerical implementation is given insection IV on page 4 and numerical examples are given in section V on page 5.

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II. Governing Equations

The motion of the body Ω(t) governed by the linear momentum equation, which can be written as follows:

ρu−∇ · σ − ρb = 0 in Ω. (1)

In the Eq. (1) upper dot represents the derivation with respect to time. Boundary and initial conditions canbe defined follows.

σ · n = g in ΓN

u = h in ΓD

(2)

u = u0 at t = 0v = v0 at t = 0

(3)

The constitutive relation between stress and strain can be written as follows for an elastic material as inEq. (4).

σ = E(x) · ε (4)

For FGMs stiffness tensor and density is a function of position. Let us define the operator D which relatesthe displacements to the strains for small strains for axi-symmetric problems.

D =

∂∂r 00 ∂

∂z1r 0∂∂z

∂∂r

(5)

Then the relation between strains and displacements can be written as follows:

ε = D · u (6)

III. Space-Time Discontinuous Galerkin Method

Let Ph(Ω) is a regular partition of the domain Ω such that Ph(Ω) is generated by division of Ω in to Ne

number of Ωe subdomains. Boundary of each subdomain Ωe is partially continuous arc for two dimensionalproblems and partially continuous surfaces for three dimensional problems. Γint defines the interelementboundaries. Let In = (tn, tn+1). Then each space-time slab can be defined as S = Ω x In.

Let us define broken space of trial functions Vn in which u(x, t) and v(x, t) are smooth and continuousvector function for every Se = Ωe x In. Similarly broken space of weighting functions Wn in which w(x, t)u

w(x, t)v are smooth and continuous vector functions for every Se = Ωe x In. Elements of Vn and Wn

are vector functions different than zero in Se and zero else where. Typical solution domain in space andspace-time slab is shown in figure 1.

(a) Solution domain (b) Space-time slab

Figure 1. Typical solution domain and space-time slab.

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Weak formulation developed in Ref. 25 is used for the discretization in space with the stabilization terms,which is developed in Ref. 26, is added. Then the bilinear and linear operators is as follows;

A(u, w) =∑

Ωe∈Ph

∫Ωe∇w · σdΩe

− ∫

Γint[w] < σ > ·n dΓint

+∫Γint

< ∇w > ·n[u]dΓint −∫ΓD

wσ · n dΓD +∫ΓD∇w · n udΓD

+µγµ

h

∫Γint

[w] · [u]dΓint +∫Γint

[λγλ

h w] · n[u] · ndΓint

+∫ΓD

µγµ

h w · udΓD +∫ΓD

λγλ

h w · n u · ndΓD

(7)

L(w) =∑

Ωe∈Ph

∫

Ωe

wρbdΩe +∑

Γe∈ΓD

∫

Γe

∇w · n hdΓe +∑

Γe∈ΓN

∫

Γe

wgdΓe. (8)

where

h =

2(

length(Γ)area(Ωe) + length(Γ)

area(Ωnb)

)−1

for Γ ⊂ Γe ∩ Γnb

area(Ωe)length(Γe) for Γ ⊂ Γe ∩ ∂Ω

.(9)

In Eq. (7) γµ and γλ are penalty parameters. Difference and averaging operators can be defined asfollows:

[φ] = (φnb − φe)

< φ >= (φnb + φe)/2

Space-time discontinuous Galerkin formulation can be written as follows based one two field formulation.16

Find u,v ∈ VnxVn such that for all wu,wv ∈ WnxWn. Then the solution of the the Eq. (10) is thesolution of the problem defined in Eq. (1) and Eq. (2).

∫In

∫Ωe

wvρvdΩedt +∫

InA(u,wv)dt− ∫

InL(wv)dt

+∫

InA(u,wu)dt− ∫

InA(v,wu)dt

+∫Ωe

wv(t+n )ρ(v(t+n )− v(t−n ))dΩe

+A(u(t+n ),wu(t+n ))−A(u(t−n ),wu(t+n ))= A(u(0+),wu(0+)) +

∫Ωe

wv(0+)ρ(v(0+)dΩe

(10)

In Eq. ( 10) wv and wu are the weight functions regarding to the momentum equation and definition ofvelocity respectively.

IV. Numerical Implementation

Numerical implementation of Eq. ( 10) is done first discretizing the momentum equation in space byusing the bilinear and linear operators which are given in Eq. ( 7) and Eq. ( 8). Then the resulting systemsof ordinary differential equations are solved by using TDG.

Bilinear interpolation functions are used for the variation of material properties in an element for FGM.

A. Space Discretization

Let u(h)(t,X) is the approximate solution of the u(t,X). Bilinear operator can be written as follows afterevaluating the integrals;

A(u(he)(t,X), w) =N∑

e=1

(Keu(t)e − Fe(u(t)e)− Fnb(u(t)nb)). (11)

In the above equation, Ke is the element stiffness matrix. Fe and Fnb arises from the surface integrals alongthe element boundaries and can be regarded as force vectors acting on the surface of the element due to theinternal and external displacements.Body forces and boundary integrals subject to boundary conditions can be written as

N∑e=1

Fb =∑

Ωe∈Ph

∫

Ωe

wρbdΩe + λ∑

Γe∈ΓD

∫

Γe

∇w · n hdΓe +∑

Γe∈ΓN

∫

Γe

wgdΓe. (12)

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Then one can write semi discrete balance equation as follows,

N∑e=1

(Meue + Keue − Fe(ue)− Fnb(unb)− Fext) = 0. (13)

In the above equations Me is the element mass matrix. The semi discrete balance equation can be writtenin a more compact form;

N∑e=1

(Meuk+1e + Keuk+1

e − Fkext) = 0. (14)

The above equation can be solved element by element by using block Gauss-Seidell method along with thetime integration method.20 In the above equation superscript k is the iteration index in the Gauss-Seidellmethod. Fext = Fb + Fnb and modified element stiffness matrix is Ke = Ke + ∂Fe

∂ue.

B. Time Integration

Time discontinuous Galerkin (TDG) is used for the time integration. In the application of TDG double fieldformulation is used. Double field formulation consist of definition of velocity and equation of motion.16

∫Iwt

([K 00 M

][uv

]+

[0 −KK 0

][uv

])dt

+wt(t+n )

[K 00 M

][u(t+n )v(t+n )

]−wt(t+n )

[K 00 M

][u(t−n )v(t−n )

]=

∫Iwt

[0F

]dt.

(15)

In equations above subscripts of the vectors and matrices are dropped for the simplicity. For the followingsection no subscript will be used to show element wise values. In Eq. ( 15) wt is the weight function in time.1st order Lagrange polynomials are used as a base function. After evaluating the integrals in Eq. ( 15), onecan obtain to the following linear equation system.

12K

12K

−13 dtK 1

6dtK−12 K 1

2K−16 dtK −1

3 dtK13dtK 1

6dtK 12M

12M

16dtK 1

3dtK −12 M 1

2M

u+n

u−n+1

v+n

v−n+1

=

Ku−n0

Fn + Mv−nFn+1

(16)

In order to decrease the computational cost of solving the set of equations defined in Eq. ( 16), algorithmTGD-B is used which is proposed by Ref. 27.

V. Results

A. One Dimensional Wave Propagation

In this section one dimensional bar suddenly loaded from its end as with a 10kPa compression load is studiedthat is shown in figure 2 on the following page. Poisson’s ratio is taken as zero. Variation of Young’s modulusand density for FGM is given in Eq. (17).Variation of material propertied along the bar is shown in figure 3on the next page. In Eq. (17) E0 = 100 MPa and ρ0 = 1 Kg/m3. Material properties for each layer ispresented in table 1. Bilinear basis functions are used. Penalty parameters γµ and γλ selected to be 3.

E(x) = E0(ax/L + 1)m

ρ(x) = ρ0(ax/L + 1)n(17)

In Eq. (17) L = 20m, a = −0.14096, m = −1.8866 and n = −3.8866.Numerical computations are made for three different time steps dt = 1e−4, dt = 5e−5 and dt = 2.5e−5

respectively. The bar is divided in to Ne = 40 and Ne = 160 elements along its length. In figure 4 on page 7numerical results obtained for different number elements and time steps is shown at point A. It is observed

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Figure 2. One dimensional bar.

x/L

E/E

0

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

1.25

1.5

1.75

2

FGMLayered

(a) Young’s Modulus (b) Density

Figure 3. Material properties along the bar.

that numerical results do not change significantly with the change of element and time step size. In case ofdt = 2.5e− 5 and Ne = 40 the numerical solution does not converges for layered and FGM bar.

In figure 5 variation of stress at different points are shown. Both FGM and layered bar solutions arein close agreement with the analytical solution obtained for FGM apart from the step changes in stress forlayered bar due to the sudden changes in material properties. On the other hand as the time proceeds thedifference between the analytical solution and numerical results increases. Thus the analytical results areobtained for a short time. It is seen that solution obtained with layered bar is more oscillatory compared tothe solution obtained from FGM bar. The amplitude of these oscillations are increasing as the time proceeds,which may lead to divergence. In addition phase difference occurs between FGM and layered bar. Also it isseen that the peak stresses are higher in FGM bar compared to the layered bar.

Variation of stress along the bar is shown in figure 6 on page 10. It is seen that changes in the stress ismore smooth in the numerical solution obtained for the FGM bar compared to the layered bar. The solutionis oscillatory in the layered bar compared to the FGM bar. It is also seen that state of the stress betweenFGM and layered bar differs significantly as time advances.

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t (sec.)

Stre

ss(k

Pa)

0 0.01 0.02 0.03 0.04 0.05-40

-30

-20

-10

0

10

20dt=1e-4,Ne=40dt=5e-5,Ne=40dt=1e-4,Ne=160dt=5e-5,Ne=160dt=2.5e-5,Ne=160

(a) FGM

t (sec.)

Stre

ss(k

Pa)

0 0.01 0.02 0.03 0.04 0.05-40

-30

-20

-10

0

10

20dt=1e-4,Ne=40dt=5e-5,Ne=40dt=1e-4,Ne=160dt=5e-5,Ne=160dt=2.5e-5,Ne=160

(b) Layered Bar

Figure 4. Results of different mesh sizes and time steps at point B; —–: FGM, - - -:Layered.

B. Suddenly Loaded Axi-Symmetric Plate

In order to make a detailed comparison axi-symmetric metal-matrix composite (MMC) plate, which is shownin figure 7, suddenly loaded from its center is studied. The thickness of the plate is h = 0.02 m and the radiusis r = 0.1 m. The load is applied up to the radius a = 0.005 m from the center. The material properties ofmetal and ceramic phase, which is shown in table 2, is taken from Ref. 4. Volume fraction of the ceramicparticles is given in Eq. ( 18) for FGM plate and in table 3 for layered plate. Rule of mixtures is used indetermining effective material properties. Layers are numbered from the bottom of the plate through thetop of the plate where the load is applied.

f = 0.6(z

0.02)0.25 (18)

Numerical computations are carried on for three different time steps which are dt = 0.2µsec., dt = 0.1µsec.and dt = 0.05µsec.. The computational domain is divided into 320, 1280, and 5120 quadratic equal sizedelements. Bilinear basis functions are used. Penalty parameters are selected as γµ = 3 and γλ = 3.

Variation of axial displacement at point z = 0 and r = 0 is shown for different time steps and elementsizes are shown in figure 8 on page 11. The numerical solution does not converges for dt = 0.05µsec. andNe = 1280 in modelling the layered plate. It is seen that grid convergence is achieved. In table 4 on thefollowing page and 5 on the next page convergence status of FGM and layered plate is presented respectively.Numerical solution converges for all time steps and element sizes. On the other hand numerical solutiondoes not converge for small time steps in modelling layered plate. The change of effective stress with timefor FGM plate at various radial and axial locations is shown in figure 9 on page 12. In figure 9 sub figuresa-d has different scaling then sub figures e-g for clarity. It is seen from the figure that the magnitude ofthe effective stress is higher at r = 0 at the points closer to the upper surface when compared to the otherradial locations. The magnitude of the effective stress decreases towards the bottom of the plate. A peak instress is seen when stress wave approaches monitoring points at r = 0. This peak vanishes as the stress wavepropagates in the radial direction. It is also seen that there are sinusoidal oscillations in the effective stressat r = 0. These oscillations are altered as the stress wave propagates in the radial direction and abrubthchanges in the effective stress is seen. These abrubth changes becomes more dominant at z/h = 0.125 andz/h = 0.25.

In figure 10 on page 13 variation of effective stress with respect to time is shown for layered plate. Similarto the figure 9, sub figures a-d has different scaling then sub figures e-g for clarity in figure 10. It is seen thatat upper part of the plate the effective stress is higher than the lower part of the plate at r = 0. There is apeak seen as the stress wave approaches to the monitoring point at the center followed by wave pattern with

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Table 1. Material properties for layered bar.

Layer 1 Layer 2 Layer 3 Layer 4Young Modulus (MPa) 103.411 110.789 119.005 128.19Density (Kg/m3) 1.07154 1.235 1.4311 1.66799

Table 2. Material properties of ceramic and metal phases of plate.

Young’s modulus (GPa) Poisson’s ratio Density (kg/m3)Metal Matrix 70 0.3 2800Ceramic 420 0.17 3200

Table 3. Volume fraction of ceramic particles inthe layers for the layered plate.

Layer 1 Layer 2 Layer 3 Layer 4f 0.200622 0.429471 0.507662 0.560675

Table 4. Convergence status for different time steps and elementsizes for FGM plate; +:converged, −:diverged.

dt = 0.2µsec. dt = 0.1µsec. dt = 0.2µsec.

dx = 2.5mm + + +dx = 1.25mm + + +dx = 0.625mm + + +

Table 5. Convergence status for different time steps and elementsizes for layered plate; +:converged, −:diverged.

dt = 0.2µsec. dt = 0.1µsec. dt = 0.2µsec.

dx = 2.5mm + − −dx = 1.25mm + + −dx = 0.625mm + + +

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t (sec.)

Str

ess

(kP

a)

0 0.005 0.01 0.015 0.02-40

-30

-20

-10

0

10

20FGMLayeredAnalytical

(a) Point A

t (sec.)

Str

ess

(kP

a)

0 0.005 0.01 0.015 0.02-40

-30

-20

-10

0

10

20FGMLayeredAnalytical

(b) Point 1

t (sec.)

Str

ess

(kP

a)

0 0.005 0.01 0.015 0.02-40

-30

-20

-10

0

10

20FGMLayeredAnalytical

(c) Point 2

t (sec.)

Str

ess

(kP

a)

0 0.005 0.01 0.015 0.02-40

-30

-20

-10

0

10

20FGMLayeredAnalytical

(d) Point B

t (sec.)

Str

ess

(kP

a)

0 0.005 0.01 0.015 0.02-40

-30

-20

-10

0

10

20FGMLayeredAnalytical

(e) Point 3

t (sec.)

Str

ess

(kP

a)

0 0.005 0.01 0.015 0.02-40

-30

-20

-10

0

10

20FGMLayeredAnalytical

(f) Point 4

Figure 5. Variation of stress on the bar at different points.

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X (m)

Str

ess

(kP

a)

0 5 10 15 20-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4t=0.01 sec.t=0.02 sec.t=0.03 sec.t=0.04 sec.t=0.05 sec.

(a) FGM

X (m)

Str

ess

(kP

a)

0 5 10 15 20-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4t=0.01 sec.t=0.02 sec.t=0.03 sec.t=0.04 sec.t=0.05 sec.

(b) Layered

Figure 6. Variation of stress along the bar at different times.

Figure 7. Axi-symmetric plate loaded from its center.

a smaller amplitude. The magnitude of the peak decreases significantly towards the bottom of the plate. Asthe stress wave propagates in the radial direction, there is a decrease in the amplitude of the effective stress.There are also abrubth changes in the effective stress at r = 0. These abrubth changes become more steeperat the center and become more regular at z/h = 0.5. These abrubth and regular changes in effecive stressvanishes at z/h = 0.125

Comparing the effective stress obtained from the FGM plate with the layered plate, one can see thatthere are sinusoidal oscillations at r = 0 in FGM plate, where there are abrubth changes in the layered plate.The average amplitude of the effective stress is very close at r = 0 for FGM and layered plate at the uppermonitoring points. On the other hand effective stress is significantly different at z/h = 0.125 and z/h = 0.25.It is seen that effective stress is higher at z/h = 0.125 and z/h = 0.25 in FGM plate than layered plate. Inaddition there are abrubth changes in the effective stress as the stress wave propagates in the radial directionin the FGM plate, where more regular pattern in effective stress is seen in the layered plate. It is also seenthat stress wave in FGM plate propagates with a higher velocity than the layered plate. The stress waveapproaches to the point z/h = 0.0 and r = 0.1 at about 3.5µsec. in FGM plate where it is 10µsec. in layered

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t (msec.)

axia

ldis

plac

emen

t(m

m)

0 0.005 0.01 0.015 0.02-0.008

-0.006

-0.004

-0.002

0

0.002

dt=2e-4 msec., Ne=320dt=1e-4 msec., Ne=320dt=5e-5 msec., Ne=320dt=2e-4 msec., Ne=1280dt=1e-4 msec., Ne=1280dt=5e-5 msec., Ne=1280dt=2e-4 msec., Ne=5120dt=1e-4 msec., Ne=5120dt=5e-5 msec., Ne=5120

(a) FGM

t (msec.)

axia

ldis

plac

emen

t(m

m)

0 0.005 0.01 0.015 0.02-0.008

-0.006

-0.004

-0.002

0

0.002dt=2e-4 msec., Ne=320dt=2e-4 msec., Ne=1280dt=1e-4 msec., Ne=1280dt=2e-4 msec., Ne=5120dt=1e-4 msec., Ne=5120dt=5e-8 msec., Ne=5120

(b) Layered

Figure 8. Variation of axial displacement at r = 0, z = 0.

plate.State of shear stress is shown in figure 11 on page 14 for two different time. There are high frequency

oscillations seen in the neighboring elements of the interface between the 1st and 2nd layers. These highfrequency oscillations propagate along the interface. These oscillations do not propagate through the interiorelements of the layers. Therefore one can conclude large change impedance causes these oscillations.

VI. Conclusion

In this study, it is observed using small time steps may lead to divergence because of the dispersioncaused by sudden change in stress at a point both for FGM ad layered media. In addition sudden changesin material properties cause dispersion in the solution which is not damped when the small time steps areused. Dispersion leads to divergence in some cases. Also certain ratio between element size and time stephas to be preserved to have accurate solutions both for one dimensional and two dimensional problems.

It is seen that using linear variation of material properties along with discontinuous Galerkin method issuccessful in modelling the mechanical behaviour of FGMs. In addition mechanical behaviour of FGM andlayered media is very similar for one dimensional problems apart from small oscillations in layered mediacaused by the sudden changes in material properties. On the other hand different mechanical behaviour isobserved between FGM and layered media for two dimensional axi-symmetric problem. It is seen that stresswaves propagates more faster in FGM media then in layered media. This shows that energy is spread toa larger volume in a short time in FGM than layered media. In addition sudden changes in stress in thelayered media may cause failure, which is not seen in FGM.

Acknowledgments

The financial support of the Scientific and Technical Research Council of Turkey to the first author isgratefully acknowledged.

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t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

5

10

15

20

25

30

35

40

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(a) z/h = 0.125

t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

5

10

15

20

25

30

35

40

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(b) z/h = 0.25

t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

5

10

15

20

25

30

35

40

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(c) z/h = 0.375

t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

5

10

15

20

25

30

35

40

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(d) z/h = 0.5

t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

20

40

60

80

100

120

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(e) z/h = 0.6255

t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

20

40

60

80

100

120

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(f) z/h = 0.75

t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

20

40

60

80

100

120

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(g) z/h = 0.875

Figure 9. Effective stress for FGM at different sections of the plate.

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t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

5

10

15

20

25

30

35

40

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(a) z/h = 0.125

t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

5

10

15

20

25

30

35

40

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(b) z/h = 0.25

t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

5

10

15

20

25

30

35

40

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(c) z/h = 0.375

t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

5

10

15

20

25

30

35

40

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(d) z/h = 0.5

t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

20

40

60

80

100

120

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(e) z/h = 0.6255

t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

20

40

60

80

100

120

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(f) z/h = 0.75

t (msec.)

Eff

ectiv

eS

tres

s(M

Pa)

0 0.005 0.01 0.015 0.020

20

40

60

80

100

120

r=0r=0.02r=0.04r=0.06r=0.08r=0.10

(g) z/h = 0.875

Figure 10. Effective stress for layered plate at different sections of the plate.

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(a) FGM; t = 5 µsec. (b) Layered; t = 5 µsec.

(c) FGM; t = 10 µsec. (d) Layered; t = 10 µsec.

Figure 11. Shear stress in the plate at two different time.

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