three-dimensional axisymmetric responses of exponentially...

Scientia Iranica A (2017) 24(3), 966{978

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Three-dimensional axisymmetric responses ofexponentially graded transversely isotropictri-materials under interfacial loading

Y. Zafaria, M. Shahmohamadia, A. Khojastehb;� and M. Rahimiana

a. School of Civil Engineering, College of Engineering, University of Tehran, Tehran, P.O. Box 11155-4563, Iran.b. School of Engineering Science, College of Engineering, University of Tehran, Tehran, P.O. Box 11155-4563, Iran.

Received 11 September 2015; received in revised form 29 January 2016; accepted 7 March 2016

KEYWORDSFunctionally gradedmaterial;Exponentially graded;Transversely isotropic;Displacementpotential;Tri-material;Axisymmetric green'sfunction.

Abstract. In this paper, an analytical formulation is presented to study an exponentiallygraded transversely isotropic tri-material under applied axisymmetric point-load and patch-load with the aid of Hankel transform and use of a potential function. The given formulationis shown to be reducible to the special cases of (1) an inhomogeneous �nite layer on a rigidbase; (2) exponentially graded bi-material or half-space under applied buried loads; (3)homogeneous tri-material or bi-material solid. Several numerical solutions are presentedto explain inhomogeneity e�ect on the stress transfer process in the inhomogeneous three-layered medium by means of a fast and accurate numerical method.

© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

Functionally Graded Materials (FGM) with continu-ous variation of mechanical properties have extensiveindustrial applications such as thermal barriers, oxi-dation, and wear-resistant coatings due to improvedresidual stress distribution, thermal properties, andlarger fracture toughness. Moreover, the better un-derstanding of anisotropic and inhomogeneous char-acteristics of rock formations and naturally depositedlayered soils is the reason for extensive studies ofstress transfer in this type of solids with broad ap-plications in geomechanics and foundation engineer-ing.

For the continuum modelling of inhomogeneity ingraded materials, di�erent types of continuous property

*. Corresponding author. Tel.: +98 21 6111 2232E-mail addresses: [email protected] (Y. Zafari);[email protected] (M. Shahmohamadi); [email protected](A. Khojasteh); [email protected] (M. Rahimian)

variations have been proposed including linear, expo-nential and power law, etc. [1].

Due to ease of formulation, for continuum model-ing of FGMs, the exponential variation of the elasticconstants is broadly used in numerical and analyt-ical analyses. For a detailed review of studies inthe �eld of anisotropic and nonhomogeneous isotropicmaterials, one might refer to [2]. The fundamentalsolutions concerning the exponentially graded elasticsolids may be found in [3-8]. Esandari and Shodja [9]obtained fundamental solutions to an exponentiallygraded transversely isotropic half-space for arbitraryburied static loads.

Kashatalyan and Rushchitisky [10] presented twodisplacement potential functions to analyze a trans-versely isotropic inhomogeneous medium with func-tionally graded Young's and Shear moduli and constantPoisson's ratio. For the analysis of wave propagationin an inhomogeneous cross-anisotropic solid, one mayrefer to [11]. Also, Eskandari-Ghadi and Amiri-Hezaveh [12] studied the wave propagation problem

Y. Zafari et al./Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 966{978 967

in an exponentially graded transversely isotropic half-space by presenting two potential functions. Selvaduraiand Katebi [13] analyzed an incompressible elastic half-space with the exponential variation of linear elasticshear modulus along the depth under vertical loads andstudied the interaction of exible and rigid plate withsuch media as [14-15].

Also, the contact analysis in graded materialsis an interesting topic. Some of recent studies ofthis type may be found in [16-20], and the Reissner-Sagoci problem for an arbitrary functionally gradedsolid in [21].

In the case of tri-materials, Kulchytsky-Zhyhailoand Bajkowski [22] studied a three-layered mediumwith two coating layers resting on half-space, where onthe top layer, the circular normal and tangential loadsare applied, and the interlayer is functionally gradedby ensuring the continuous change of Young's modulusbetween the layers.

In this paper, with the aid of displacement po-tential functions presented by Eskandari-Ghadi andAmiri-Hezaveh [12], under axisymmetric condition, anexponentially graded transversely isotropic tri-materialis analyzed for di�erent types of vertical point, patch,and ring load patterns. Green's functions of stressand displacement components are given in terms ofexplicit line-integral representations. The results areveri�ed to be in exact agreement with the solutionsof inhomogeneous transversely isotropic half-space,homogeneous transversely isotropic tri-material full-space, and incompressible inhomogeneous isotropichalf-space [9,23,13], respectively. Also, in the case ofhomogeneous transversely isotropic �nite layer on arigid base, the results are con�rmed with the studyby Small and Booker [24].

In this paper, the results may be utilized forthe boundary integral equation formulation of geome-chanical and foundation engineering problems and theinterfacial fracture analysis in composite materials, orthey may be used as benchmark solutions for thedevelopment of numerical methods for the analysis ofinhomogeneous anisotropic materials.

2. Statement of the problem and thegoverning equation

With reference to Figure 1, a tri-material solid, con-sisting of a �nite layer of arbitrary thickness (h)perfectly bonded to two half-spaces, is considered for acylindrical coordinate system (r; �; z). All layers areassumed to be composed of di�erent linearly elastictransversely isotropic materials where z-axis is thecommon axis of symmetry of materials, which is normalto the horizontal interfaces of the domains. Upperhalf-space (z < 0) is denoted as Layer I, middle �nitelayer (0 < z < h) as Layer II, and the lower half-

Figure 1. Functionally graded transversely isotropictri-material under arbitrary interfacial vertical load.

space (z > h) as Layer III. The elasticity constantsof the layers are designated as CIij(z), CIIij (z), andCIIIij (z), where the superscripts I, II, and III indicatethe properties of Layers I, II, and III, respectively.The inhomogeneity is considered with the arbitraryexponential variation of material properties along z-axis in each layer. Therefore, the elasticity constantsare expressed as:

in Layer I:

cIij(z) = CIije

2�Iz; (1)

in Layer II:

cIIij (z) = CIIij e

2�IIz; (2)

in Layer III:

cIIIij (z) = CIIIij e

2�IIIz; (3)

where CIij , CIIij , and CIIIij are the elasticity constantscorresponding to depths z = 0�, z = 0+, and z = h+,respectively, and relation C66 = (C11 � C12)=2 existsbetween the constants. Term � is the exponentialfactor characterizing the alteration made to materialproperties along z where � = 0, representing a ho-mogenous transversely isotropic tri-material.

For an exponentially graded transversely isotropicelastic material, in the absence of body forces andaxisymmetric static condition, the governing equilib-rium equations in terms of displacement componentsand cylindrical coordinate system may be expressedas:

968 Y. Zafari et al./Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 966{978

C11 =�@2ur@r2

+1r@ur@r� urr2

�+ C44

@2ur@z2

+ (C13 + C44)@2uz@r@z

+2�C44�@ur@z

+@uz@r

�=0

C44�@2uz@r2

+1r@uz@r

�+ C33

@2uz@z2

+ (C13 + C44)�@2ur@r@z

+1r@ur@z

�2�C33

@uz@z

+ 2�C13�@ur@r

+urr

�= 0; (4)

where ur and uz are the displacement components inradial (r) and vertical (z) directions, respectively. Inorder to uncouple the above equilibrium equations,displacement potential function (F ), introduced in[12], is used. This potential function is related todisplacement components ur and uz as:

ur(r; z) = ��3 @2F (r; z)@r@z

� 2�2� @F (r; z)@ruz(r; z) =

�(1 + �1)r2r + �2 @@z

�2� +

@@z

��F (r; z);

(5)

where:

r2r = @2

@r2+

1r@@r; (6)

and:

�1 =C12 + C66

C66; �2 =

C44C66

;

�3 =C13 + C44

C66; �4 =

C33C66

: (7)

Putting Eq. (5) into Eq. (4) leads to a single partialdi�erential equation as the governing equation forpotential function, F :�r21r22 � 4�3 � �21 + �1 �

2r2r�F (r; z) = 0; (8)

where:

r2i = r2r + 1s2i�@2

@z2

�� 2s2i; i = 1; 2 (9)

�@2

@z2

��Y =

@2

@z2(e�zY ): (10)

Here, s1 and s2 are the roots of the following equationand are not zero or pure imaginary numbers [25]:

�4�2s4+��23��22�(1+�1)�4� s2+�2(1+�1)=0:

(11)

The zeroth-order Hankel transform is de�ned as [26]:

~f0(�; z) =Z 1

0rf(r; z)J0(r�)dr;

f(r; z) =Z 1

0

g�f0(r; z)J0(r; �)d�; (12)where J0 is the Bessel function of the �rst kind oforder zero. By applying such a transform to Eq. (8)with respect to the radial coordinate, the followingordinary di�erential equation is obtained for F :�

~r201 ~r202 + 4�3 � �21 + �1 �2�2�

~F 0 = 0; (13)

where:

~r20i = ��2 + 1s2i�d2

dz2

�� 2s2i; i = 1; 2: (14)

By means of Eq. (5) and the identities involv-ing Hankel transforms, the transformed displacement-potential relations may be written compactly as:

~u1r = �3�@ ~F 0(�; z)

@z+ 2�2�� ~F 0

~u0z =��2

@@z

(2� +@@z

)� �2(1 + �1)�

~F 0: (15)

The similar expressions for stress-potential functionsmay be expressed as:

~�1zr =�C44�

�(�3 � �2) @

2

@z2+ �2(1 + �1)

�~F 0�e2�z;

~�0zz =�C33

@@z

��2

@@z

(2� +@@z

)� �2(1 + �1)

+ �3C13C33

�2�

~F 0 + 2�2�C13�2 ~F 0�e2�z;

~�0rr + 2C66e2�z

(�~urr

�0)= e2�z

�@@z

��3C11�2

� C13�2(1 + �1) + �2C13 @@z�

2� +@@z

��+ 2�2�C11�2

�~F 0;

~�0�� 2C66e2�z(�

~urr

�0)= e2�z

�@@z

��3C12�2


� C13�2(1 + �1) + �2C13 @@z�

2� +@@z

��+ 2�2�C12�2

�~F 0: (16)

The general solution to Eq. (13) can be written as:

~F 0(�; z) =e��z�A(�)e�1z +B(�)e��1z + C(�)e�2z

+D(�)e��2z�; (17)

where constants A; :::;D are determined by satisfyingthe boundary conditions. Also, �1 and �2 are obtainedas:

�1 =ra�2 + b+

12pc�4 + d�2;

�2 =ra�2 + b� 1

2pc�4 + d�2; (18)

where:

a =12

(s21 + s22); b = �

2;

c = (s21 � s22)2; d = �16�3 � �2�4 �2: (19)

Consistent with the regularity condition at in�nity, thegeneral solution to F , Eq. (17), can be rearranged as:

in Layer I:

~F 0(�; z) = e��IzhAI(�)e�

I1z + CI(�)e�

I2zi; (20)

in Layer II:

~F 0(�; z) =e��IIz�AII(�)e�

II1 z +BII(�)e��II1 z

+ CII(�)e�II2 z +DII(�)e��II2 z

�; (21)

in Layer III:

~F 0(�; z) =e��IIIz�BIII(�)e��III1 z

+DIII(�)e��III2 z�: (22)

Therefore, determination of the above 8 coe�cientsfor the exponentially graded tri-material full-spaceproblem of interest is required.

With the aid of equations Eqs. (15) and (16), theimposition interfacial stress and displacement compat-ibility conditions associated with a tri-material FGMunder applied load are greatly facilitated, as illustratedin the following sections.

3. Applying boundary conditions

3.1. Functionally graded tri-material underloading at z = 0 or z = h

An axisymmetric vertical load on subdomain �0 or �h,located at depth z = 0 or z = h, respectively, may bewritten as a prescribed stress discontinuity:

�zz(r; 0�)� �zz(r; 0+) =(�R(r); r 2 �0

0; r =2 �0�rz(r; 0�) = �rz(r; 0+); 0 � r � 1;

or:

�zz(r; h�)� �zz(r; h+) =(�R(r); r 2 �h

0; r =2 �h�rz(r; h�) = �rz(r; h+); 0 � r � 1; (23)

where R(r) is the speci�ed interfacial traction invertical direction. Eight equations are obtained to�nd unknown coe�cients AI ; :::; DIII with respect tothe aid of displacement, stress potential functions(Eqs. (15) and (16)) and Relations (20) to (22),stress discontinuity condition (Eq. (23)) along with thecontinuity of stresses, and displacements along z = 0and z = h. These equations are arranged in matrixform as:

I(�)�AI(�) CI(�) AII(�) BII(�) CII(�) DII(�)

BIII(�) DIII(�)�T

= S0(�); (24)

where I(�) is given in the Appendix, and S0(�) is thesource vector described by:

S0(�) =�0 0 0 0 0 �R0(�) 0 0�T ; (25)

for loading at z = 0 and:

S0(�) =�0 0 0 0 0 0 0 �R0(�)�T ; (26)

for loading at z = h. The term R0(�) is theHankel transformed function of the applied traction.Setting �I = �II = �III = 0, the above formulationdegenerates exactly to the static analysis of an axiallyloaded homogeneous transversely isotropic tri-materialby Khojasteh et al. [23].

3.2. Functionally graded �nite layer on a rigidbase under surface loading at z = 0

In the previously described analysis, the formulation ofan exponentially graded �nite layer on a rigid base isobtained (see Figure 2) by removing Layer I, CIij ! 0,


Figure 2. Functionally graded �nite layer on a rigidhalf-space due under applied patch-load with a rough-rigidcondition at interface.

applying load to the top of Layer II, and extremelyincreasing the elastic constants of Layer III to resemblea rigid half-space. The stress boundary conditions onthe surface of layer, z = 0, are described as:

�zr(r; 0) = 0

�zz(r; 0) =

(�R(r); r 2 �00; r =2 �0 (27)

Here, the boundary conditions on the lower interfaceare assumed to be the rough-rigid or smooth-rigid [27]cases, de�ned as fully bonded or free sliding conditions,respectively.

The rough-rigid case which resembles the in�nitefriction capacity at the interface is de�ned as:

ur(r; h) = uz(r; h) = 0; (28)

and the smooth-rigid condition may be described as:

uz(r; h) = 0; �zr(r; h) = 0: (29)

By applying these boundary conditions, the matrixformation of the equations is given as:

I(�)�AII(�) BII(�) CII(�) DII(�)

�T= S0(�); (30)

where I(�) is presented in Appendix A, and S0(�) iswritten as:

S0(�) =�0 �R0(�) 0 0�T : (31)

4. Axisymmetric Green's functions

In this section, the axisymmetric Green's functions fordi�erent types of loading are given.

4.1. Point-loadA vertical point load with resultant Fv and the relevantHankel transformed loading coe�cient can be written

as:

R(r) =Fv�(r)

2�r! R0(�) = Fv

2�: (32)

With the aid of inverse Hankel transform and usingEqs. (15) and (16), the displacement and stress re-sponses of solid due to vertical point-load may beexpressed as:

ûzr(r; z;�) =Z 1

0�2��3d ~F 0

@z+ 2�2� ~F 0

�J1(r�)d�

ûzz(r; z;�) =Z 1

0��2

ddz

(2� +ddz

)� �2(1 + �1)�

~F 0J0(r�)d�; (33)

and:

�̂zzr(r; z;�) =e2�zZ 1

0�2�C44�

(�3 � �2) d2

dz2

+ �2(1 + �1)�

~F 0�J1(r�)d�;

�̂zzz(r; z;�) =e2�zZ 1

0��C33

ddz

��2

d@z

(2� +ddz

)

� �2(1 + �1) + �3C13C33 �2�

+ 2�2�C13�2�

~F 0J0(r�)d�;

�̂zrr(r; z;�) +2e2�zC66

rûzr = e

2�zZ 1

0��ddz

�C11�3�2

� C13�2(1 + �1) + �2C13 ddz (2� +ddz

)�

+ 2�2�C11�2�

~F 0J0(r�)d�;

�̂z��(r; z;�)�2e2�zC66r

ûzr = e2�zZ 1

0��ddz

�C12�3�2 � C13�2(1 + �1)

+ �2C13ddz

(2� +ddz

)�

+ 2�2�C12�2�

~F 0J0(r�)d�; (34)

where symbols ûzi and �̂zik (i; k = r; z) indicate thedisplacement and stress Green's functions, respectively,and superscript z indicates the direction of appliedpoint-load.


4.2. Patch-loadFor a vertical circular patch-load of radius a andresultant Fv, displacement and stress Green's functionsmay be obtained by Eqs. (33) and (34) with thefollowing relation for loading function:

R(r) =Fv�a2

! R0(�) = Fv J1(a�)��a : (35)4.3. Ring-loadSimilar to the patch-load analysis, the displacementand stress responses under the action of a vertical ring-load acting at radius a and resultant Fv are given byEqs. (33) and (34) with the following adaptation forloading function:

R(r) = Fv�(r � a) ! R0(�) = FvJ0(a�)2� : (36)

5. Special cases

By means of the previously obtained solution for an ex-ponentially graded transversely isotropic tri-material,some special cases of interest may be degenerated asfollows.

5.1. Exponentially graded transverselyisotropic bi-material

To obtain the formulation of an exponentially gradedtransversely isotropic bi-material, one may set h =0, which means omitting the middle �nite layer orsubstituting CIij = CIIij and �I = �II . In this case,by setting �I = �II = 0, the static Green's functionsof a homogeneous transversely isotropic bi-material areobtained as in [28].

5.2. Homogeneous transversely isotropictri-materials

Upon setting �I = �II = �III = 0, the solutions forhomogeneous transversely isotropic tri-materials areobtained as in [23] in static condition.

5.3. Exponentially graded half-spaceConsidering CIij ! 0, �II = �III = �, and CIIij =CIIIij = Cij , under the action of surface or buriedloads, Green's functions of an exponentially gradedtransversely isotropic half-space [9] or incompressibleisotropic solid [13] are obtained.

5.4. Homogeneous �nite layer on a rigidhalf-space

By applying load at z = 0 and setting CIij ! 0 andCIIIij ! 1, the response of a functionally graded�nite layer on a rigid base is obtained which is exactlyidentical to the results in [24] and [27] for a �nite layeron a rough-rigid half-space.

6. Numerical evaluation

In the previous section, the axisymmetric Green's

functions associated with di�erent load con�gurationshave been expressed in terms of one-dimensional semi-in�nite integrals. Since the closed-form integrationis not possible [29-33], a numerical quadrature tech-nique has to be adopted. For such evaluations, itis essential to consider the oscillatory nature of theintegration due to the presence of Bessel functions andthe singular nature of stress transfer. Here, an adaptivequadrature rule has been utilized successfully by usingWolfram Mathematica software. For veri�cation ofthe present study with the existing numerical solutionsin the literature, three comparisons are made. First,for a homogeneous tri-material, vertical displacementof the medium due to point-load is determined andcompared with the results given by Khojasteh etal. [23]. As shown in Figure 3, the solutions areidentical. Also, under applied buried vertical patch-load, the axial displacement along the depth of anexponentially graded half-space for di�erent inhomo-geneity parameters (�) is compared with the resultsby Eskandari and Shodja [9] (Figure 4). Here, it isnoteworthy that for an isotropic material, the elasticityconstants may be reduced to C11 = C33 = � + 2�,C12 = C13 = �, and C44 = C66 = � where �and � are the Lame's constants of the isotropic solid.For an incompressible isotropic solid with the initialshear modulus of �0 = 3:33, under applied patch-load, vertical displacement along z-axis is shown inFigure 5 and is compared with the results obtainedby Selavadurai and Katebi [13], which shows the goodagreement of results.

Figure 3. Displacement Green's function ûzz alongz-direction due to point-load compared with the result byKhojasteh et al. [23] for homogenous materials (�L = 0).


Figure 4. Displacement Green's function ûzz along z-axiscompared with the study by Eskandari and Shodja [9] forfunctionally graded transversely isotropic half space.

Figure 5. Displacement Green's function ûzz along r-axiscompared with the results by Selvadurai and Katebi [13]for incompressible isotropic half-space.

In this study, three di�erent synthetic trans-versely isotropic materials are considered for construct-ing the parametric study of analysis results as listed inTable 1.

In the following, three di�erent material con�gu-rations in layers are considered:

Table 1. Elastic constants of transversely isotropicmaterials.

Materialno.

C11(Gpa)

C12(Gpa)

C13(Gpa)

C33(Gpa)

C44(Gpa)

Material 1 5.6 1.6 1.8 10.9 2Material 2 5.5 1.5 1.8 15.9 2Material 3 5.4 1.4 1.7 25.9 2

Figure 6. Displacement Green's function ûzz along z-axisdue to unit point-load (�L = 0:1).

- Case 1: Material 2 in all three layers;

- Case 2: Materials 1, 2, and 3 in Layers I, II, and III,respectively;

- Case 3: Similar to Case 2, Layer 1 is omitted, CIij !0, which resembles the two-layered half-space or maybe used for buried-load con�guration analysis.

By considering the described cases, Figures 6and 7 depict displacement Green's function ûzz alongthe depth of point-load with � � L = 0:1, thickness ofmiddle layer h = 2L, and patch-load with the resultantof 1=�, � � a = 0:1 and h = 2a, respectively, whereL is the unit of length. Based on the �gures, oneshould notice the continuity at interface as well asthe singularity of the response at point load locationaccompanying the signi�cant e�ect of inhomogeneityparameter.

Figures 8 and 9 depict the inuence of thickness ofmiddle layer, h, on the displacement Green's functionûzz along the depth and radial directions, respectively.For higher thickness of middle layer, one may observethe lower displacement response which is mainly the


Figure 7. Displacement Green's function ûzz along z-axisdue to patch-load with resultant 1=�(�a = 0:1).

Figure 8. Displacement Green's function ûzz alongz-direction due to point-load for di�erent middle layerthicknesses (�a = 0:1).

result of exponential increase of elasticity constant ofthe layer.

To survey the e�ect of inhomogeneity parameteronly on the middle layer in Case 1, the inhomogeneityparameter has been changed in Figures 10 to 12, with� = 0 representing the homogeneous layer. Figure 10

depicts vertical displacement ûzz of the medium underapplied point-load along depth, which shows the lower

Figure 9. Displacement Green's function ûzz alongr-direction due to patch-load for di�erent middle layerthicknesses (�a = 0:1).

Figure 10. Displacement Green's function ûzz alongz-axis due to unit point-load.

displacement in all three layers of higher inhomogeneityparameter � as a result of higher sti�ness of the middlelayer. A similar trend is observed for the response inFigure 11 for displacement under patch-load. For thesustained vertical stress in the medium, Figure 12 ispresented which shows a more rapid stress transfer bythe increase of �.

For studying the response of an inhomogeneous�nite layer on a rigid base, Figures 13-17 are included.In these �gures, Layer I is omitted (i.e., CIij ! 0),Material 1 is used in Layer II, and the material param-eters are highly increased for Layer III to resemble therigid base layer (i.e., CIIIij ! 1). Also, the boundaryconditions on the interface of the two layers are de�ned


Figure 11. Displacement Green's function ûzz alongz-axis due to patch-load with resultant 1=�.

Figure 12. Stress Green's function �̂zzz along z-axis dueto patch-load with resultant 1=�.

as rough and smooth, which means an interface within�nite and without friction capacity, respectively.

Figure 13 depicts the vertical stress distributiondue to patch-load in a transversely isotropic �nite layerbonded to a rigid base, compared with the solutions bySmall and Booker [24]. Figures 14 and 15 illustratethe inuence of inhomogeneity parameter on verticalstress distribution (�̂zzz) due to a patch-load in verticaldirection and rough/smooth-rigid conditions, respec-

Figure 13. Stress Green's function �̂zzz beneath the edgeof a circular loading along z-direction for �nite layer on arigid half-space with rough-rigid conditions compared withthe results by Small and Booker [24].

Figure 14. Stress Green's function �̂zzz along z-axis dueto patch-load with resultant 1=� for a �nite layer on arigid half-space with rough-rigid condition.

tively. Based on the �gures, one should notice thesigni�cant e�ect of inhomogeneity as in sti�er medium,i.e. the higher inhomogeneity factor is, the higher stressreaches the interface. The inuence of inhomogeneityon vertical displacement of solid (ûzz) is shown in Fig-ure 16. From the �gure, one should notice the decreaseof displacement along the depth and e�ect of roughcondition on the decrease of displacement, especiallyfor lower �. Finally, Figure 17 represents the radialdisplacement component (ûzr) along the radial directionat the interface with the smooth-rigid condition. Fromthe �gure, one should notice the signi�cant reductionof displacement as the e�ect of higher sti�ness of the


Figure 15. Stress Green's function �̂zzz along z-axis dueto patch-load with resultant 1=� for a �nite layer on arigid half-space with smooth-rigid condition.

Figure 16. Displacement Green's function ûzz alongz-axis due to patch-load with resultant 1=� for a �nitelayer with rough-rigid and smooth-rigid conditions.

Figure 17. Displacement Green's function ûzr alongr-axis direction due to the patch-load and for a �nite layerwith smooth-rigid conditions.

layer. It is trivial that ûzr at interface tends to zero forthe case of rough-rigid condition.

7. Conclusion

The axisymmetric response of an exponentially gradedtransversely isotropic tri-material elastic solid due topoint-load and patch-load in static condition is pre-sented in this study. By use of one displacementpotential function and Hankel transform, di�erentcomponents of response of the medium under appliedloads are given in terms of in�nite line-integrals. Theseequations are useful for e�ective boundary element for-mulations of elastostatic problems and mixed boundaryvalue problem formulations in inhomogeneous andanisotropic composites. These Green's functions can bedegenerated analytically and numerically for di�erentcases of exponentially graded transversely isotropic bi-material, homogeneous full-space tri-material, expo-nentially graded half space, and special case of �nitelayer on a rigid half-space. Numerical examples showthe signi�cant inuence of the thickness of middle layerand the degree of inhomogeneity of the material on allaspects of the response.

Nomenclature

a Loading radiusAm; :::; Dm Constants of integrationCij Elasticity constantsF Potential functionFGM Functionally Graded MaterialFv Vertical point-load resultant valueh Middle layer thicknessJm Bessel function of the �rst kind and

m-th orderm Hankel integral transformr Radial coordinateR(r) Speci�ed interfacial traction in z-axis

directions1; s2 Roots of strain energy functionur Displacement components in r-

directionsuz Displacement components in z-

directionsz0 Transformed loading coe�cient� Exponential factor characterizing the

degree of material properties gradient�0 Arbitrary area�h Arbitrary area�ij Stress components� Lambs constant� Lambs constant


References

1. Birman, V. and Byrd, L.W. \Modeling and analysis offunctionally graded materials and structures" , AMR.ASME., 60(50), pp. 195-216 (2007).

2. Wang, C.D., Tzeng, C.S., Pan, E. and Liao, J.J.\Displacements and stresses due to a vertical pointload in an inhomogeneous transversely isotropic half-space" , Int. J. Rock Mech. Min., 40(5), pp. 667-685(2003).

3. Wang, C.D., Pan, E., Tzeng, C.S., Han, F. andLiao, J.J. \Displacements and stresses due to a uni-form vertical circular load in an inhomogeneous cross-anisotropic half-space" , Int. J. of Geomechanics.,6(1), p. 110 (2006).

4. Wang, C.D. and Tzeng, C.S. \Displacements andstresses due to non-uniform circular loadings in an in-homogeneous cross-anisotropic material", Mech. Res.Commun., 36, pp. 921-932 (2009).

5. Martin, P.A., Richardson, J.D., Gray, L.J. and Berger,J.R. \On Green's function for a three-dimensionalexponentially graded elastic solid" , Pro. R. Soc.London, Ser. A., 458, pp. 1931-1947 (2002).

6. Pan, E. and Yang, B. \Three-dimensional interfacialGreen's functions in anisotropic bi-materials" , Appl.Math. Model., 27, pp. 307-326 (2003).

7. Chan, Y.S., Grey, L.J., Kaplan, T. and Paulino, G.H.\Green's function for a two-dimensional exponentiallygraded elastic medium" , Pro. R. Soc. London, Ser.A., pp. 1689-1706 (2004).

8. Sallah, O.M., Gray, L.J., Amer, M.A. and Matbuly,M.S. \Green's function expansion for exponentiallygraded elasticity" , Int. J. Numer. Meth. Eng., 82,pp. 756-722 (2010)

9. Eskandari, M. and Shodja, H.M. \Green's functionsof an exponentially graded transversely isotropic half-space" , Int. J. of Solids Struct., 47, pp. 1537-1545(2010).

10. Kashtalyan, M. and Rushchitsky, J.J. \Revisitingdisplacement functions in three-dimensional elasticityof inhomogeneous media" , Int. J. of Solids Struct.,46, pp. 3463-3470 (2009).

11. Wang, C.D., Lin, Y.T., Jeng, Y.S. and RuanZ.W. \Wave propagation in an inhomogeneous cross-anisotropic medium", Int. J. for NAMG., 34, p. n/a-n/a (2009).

12. Eskandari-Ghadi, M. and Amiri-Hezaveh, A. \Wavepropagations in exponentially graded transverselyisotropic half-space with potential function method",Mechanics of Materials., 68, pp. 275-292 (2014).

13. Selvadorai, A.P.S. and Katebi, A. \Mindlin's problemfor an incompressible elastic half-space with an expo-nential variation in the linear elastic shear modulus",Int. J. ES., 65, pp. 9-21 (2013).

14. Katebi, A. and Selvadurai, A.P.S. \A frictionlesscontact problem for a exible circular plate and anincompressible non-homogeneous elastic halfspace",Int. J of Mech. Sci., 90, pp. 239-245 (2015).

15. Selvadurai, A.P.S. and Katebi, A. \An adhesive con-tact problem for an incompressible non-homogeneouselastic halfspace", Int. J of Acta Mechanica, 226(2),pp. 249-265 (2015).

16. Ke, L.-L. and Wang, Y.-S. \Two-dimensional slidingfrictional contact of functionally graded materials",European J. of Mechanics - A/Solids., 26(1), pp. 171-188 (2007).

17. Liu, T., Wang, Y. and Xing, Y. \Fretting contact oftwo elastic solids with graded coatings under torsion",Int. J. of Solids Struct., 49(10), pp. 1283-1293 (2012).

18. Chen, P. and Chen, S. \Partial slip contact betweena rigid punch with an arbitrary tip-shape and anelastic graded solid with a �nite thickness", Int. J. ofMechanics of Materials, 59, pp. 24-35 (2013).

19. Volkov, S., Aizikovich, S., Wang, Y.S. and Fedotov, I.\Analytical solution of axisymmetric contact problemabout indentation of a circular indenter into a softfunctionally graded elastic layer" , Int. J. of Acta.Mechanica, 29(2), pp. 196-201 (2013).

20. Liu, T.-J., Xing, Y.-M. and Wang, Y.-S. \The axisym-metric contact problem of a coating/substrate systemwith a graded interfacial layer under a rigid sphericalpunch" , Int. J. of Mathematics and Mechanics ofSolids, 21(3), pp. 383-399 (2014).

21. Matysiak, S.J., Kulchytsky-Zhyhailo, R. andPerkowski, D.M. \Reissner-Sagoci problem for ahomogeneous coating on a functionally graded half-space", Int. J. of Mechanics Research Communi-cations, 38(4), pp. 320-325 (2011).

22. Kulchytsky-Zhyhailo, R. and Bajkowski, A.S.\Stresses in coating with gradient interlayer causedby contact loading", Int. J. of Acta Mechanica etAutomatica, 8(1), pp. 33-37 (2014).

23. Khojasteh, A., Rahimian, M. and Eskandari, M.\Three-dimensional dynamic Green's functions intransversely isotropic tri-materials", Appl. Math.Model., 37, pp. 3164-3180 (2013).

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�IIIi =CIII44

��2�III2 �III(�III + �IIIi )

+ �III3 (�III + �IIIi )

2 � !IIIi�;

�Ii = (BI + �Ii )C

I33#

Ii + C

I13�

2'Ii ;

�IIi = �(BII � �IIi )CII33#IIi + CII13 �2'IIi ;�IIi = �(BII + �IIi )CII33!IIi + CII13 �2�IIi ;�IIIi = �(BIII + �IIIi )CIII33 !IIIi

+CIII13 �2�IIIi ;

�IIi = e�(�II��IIi )h'IIi ;

�IIi = e�(�II+�IIi )h�IIi ;

�IIIi = e�(�III��IIIi )h�IIIi ;

IIi = e�(�II��IIi )h#IIi ;

IIi = e�(�II+�IIi )h!IIi ;

IIIi = e�(�III+�IIIi )h!IIIi ;

pIIi = e(�II+�IIi )h�IIi ;

qIIi = e(�II��IIi )h�IIi ;

qIIIi = e(�III��IIIi )h�IIIi ;

nIIi = e(�II+�IIi )h�IIi ;

tIIi = e(�II��IIi )h�IIi ;

tIIIi = e(�III��IIIi )h�IIIi : (A.4)

Biographies

Yaser Zafari received his MSc degree from Schoolof Civil Engineering, University of Tehran, Iran, in2016. He has studied mechanics of composites andinhomogeneous materials through analytical and nu-merical methods. His research �elds are computationalmechanics, elastostatics of anisotropic materials andcomposite structures.

Mehdi Shahmohamadi received his PhD degreefrom University of Tehran, Iran, in 2017. He hasstudied interaction analysis in anisotropic and inho-mogeneous materials through semi-analytical methods.He has published 7 papers in ISI indexed journals.

Ali Khojasteh, the current Assistant Professor atthe University of Tehran, received his MSc and PhDdegrees from the University of Tehran, Iran, in 2005and 2009, respectively, both ranked 1st among grad-uates. His research interests are wave propagationin solids/uids, computational mechanics, mechanicsof composites, boundary element method, and soil-structure Interaction. He was ranked 262nd amongmore than 800,000 and 7th among more than 15000participating in the nationwide undergraduate andMSc program entrance exams, respectively. He wasalso ranked 9th among more than 15000 in the nationalCivil Engineering Olympiad. He has published morethan 20 papers in international ISI indexed journals.

Mohammad Rahimian, currently the Full Professorof Civil Engineering Department at the University ofTehran, received his BSc in Civil Engineering fromUniversity of Tehran. He received his PhD in EcoleNational Des Ponts et Chaussees, France, in 1981.His research �elds are earthquake engineering, elasto-dynamics of anisotropic materials, soil-structure-uidinteraction, etc. He is the Assistant to Chair of theUniversity of Tehran and the Former Chair of theUniversity of Tehran. More than 45 papers published ininternational ISI indexed journals, along with 6 writtenor translated books are of his scienti�c works.

three-dimensional axisymmetric responses of exponentially...

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