Mechanical analysis of a functionally graded cylinder...

Scientia Iranica B (2015) 22(2), 493{503

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Mechanical analysis of a functionally gradedcylinder-piston under internal pressure due to acombustion engine using a cylindrical super elementand considering thermal loading

R. Pourhamida, M.T. Ahmadianb;�, H. Mahdavy Moghaddama andA.R. Mohammadzadeha

a. Department of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran.b. School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran.

Received 27 April 2014; received in revised form 22 July 2014; accepted 17 November 2014

KEYWORDSSuper ElementMethod (SEM);Mechanical analysisand thermal loading;Functionally gradedmaterial;Cylinder;Piston.

Abstract. Increasing the performance of combustion engines has always been ofinterest to engine designers. In this regard, one approach has been to implement newlydeveloped materials. Functionally Graded Materials (FGMs) have been shown to be heattreatable materials for high-temperature environments with thermal protection. Usingthese materials requires an understanding of the development of temperature and stressdistribution in the transient state. This information is considered to be the main key inthe design and optimization of devices for failure prevention. In this paper, a cylindricalSuper Element (SE) has been employed to investigate the mechanical stress distribution of acylinder-piston mechanism made of FGM for a combustion engine design. Analysis indicatesthat modeling a cylinder and piston with a few super elements gives the same results asa conventional �nite element with 10052 elements. Materials are selected to be ceramicsand metal with power low distribution. Results indicate that temperature distribution inthe thickness of the cylinder does not vary, as the power of material distribution exceedsthe value of one, while temperature distribution variation is sensitive to power distributionless than one. This has a high e�ect on stress distribution in both FG cylinders and FGpistons.© 2015 Sharif University of Technology. All rights reserved.

1. Introduction

Functionally Graded Materials (FGMs) are modern in-homogeneous materials in which two or more di�erentmaterial ingredients change continuously and graduallyresulting in elimination of interface problems and,consequently, uniformity of stress and temperaturedistribution. Frequently, these materials are made from

*. Corresponding author. Tel.: +98 21 66165503;Fax: +98 21 66000021E-mail address: [email protected] (M.T. Ahmadian)

ceramic and metal. Usually, the composition variesfrom a ceramic-rich surface to a metal-rich surface.This behavior is according to a desired variation ofthe volume fraction. These changes are consideredas power or exponential functions in the radius orthickness directions [1-3].

This material can be used in various industrialapplications, such as engine components, especially incylinder blocks and pistons. The application of FGMscan be an inventive phenomenon in internal combustionengines [4], but, until now, FGM has not been used inthe industry of internal combustion engines. However,

494 R. Pourhamid et al./Scientia Iranica, Transactions B: Mechanical Engineering 22 (2015) 493{503

if we consider the thermal barrier coating system asa type of FGM, they were used in Cummins engines,commercially [5].

The FGM structure with ceramic is usually pre-ferred in the design of pistons and cylinders because ofthe preference in thermal barrier and wear resistance.In recent years, scientists have presented several piecesof research on thermal and stress analyses of FGMs,using semi-analytical and numerical methods. Are� [6]presented the nonlinear thermo-elastic analysis of athick-walled functionally graded piezoelectric cylinderunder thermal, mechanical and electrical loads. Heproposed a novel analytical method for estimating theresponse of systems of nonlinear di�erential equationsbased on the modi�cation and combination of Ado-mian's decomposition and successive approximationmethods.

Dai and Rao [7] studied the dynamic thermo-elastic behavior of a double-layered hollow cylinderwith an FGM layer. They utilized the �nite di�er-ence method and the Newmark method to solve theproblem under both dynamic mechanical and thermalloads. Feng et al. [8] presented the thermo-mechanicalanalysis of FG cylindrical vessels, using an edge-based smoothed Finite Element Method (FEM). In thismethod, the problem domain was �rstly discretizedinto triangular elements, and edge-based smoothing do-mains were further formed along the edges of triangularmeshes. In order to improve the accuracy, sti�nessmatrices were calculated using a strain smoothingtechnique in the smoothing domains. Nie and Batra [9-10] presented material analysis of FG isotropic andincompressible linear elastic hollow cylinders. Theyused the Airy stress function to derive exact solutionsfor plane strain deformations.

Safari et al. [11] conducted two-dimensional dy-namic analysis of thermal stresses in a �nite-lengthFG thick hollow cylinder, subjected to thermal shockloading. Using the Laplace transform and the seriessolution, thermo-elastic Navier equations in the dis-placement form were solved analytically. Desai andKant [12] presented a mixed semi-analytical solutionfor FG �nite length cylinders of orthotropic materials,subjected to thermal loads. In this research, boundaryconditions were satis�ed exactly by taking an analyticalexpression in terms of the Fourier series expansion.First order simultaneous ordinary di�erential equationswere obtained as the mathematical model, which wereintegrated through an e�ective numerical integrationtechnique by �rst transforming the boundary valueproblem into a set of initial value problems. Asemi etal. [13] carried out the dynamic analysis of thick shortlength FG cylinders. In this article, the Finite ElementMethod (FEM), based on the Rayleigh-Ritz energyformulation, was applied. Besides, the Newmarkdirect integration method was also used to solve time

dependent equations. Darabseh et al. [14] presentedthe transient thermo-elasticity analysis of a FG thickhollow cylinder, based on the Green-Lindsay model.The heat conduction equation and the equation ofmotion were solved using the Galerkin �nite elementmethod. Investigating the thermal analysis of FGcylinders and FG pistons in the application of FGMsin internal combustion engines is very rare. Akhlaghiet al. [15] studied heat loss and thermal stress indirect injection diesel engines, using FGMs. They useda combination of �nite element and �nite di�erencemethods.

Due to the high computational time of the con-ventional �nite element method, researchers have triedto develop super elements to overcome this problem.Koko and Olson [16] developed a super element toanalyze stress distribution in a plate. Ahmadian andZangane [17] used a super element to perform thefree vibration analysis of laminated sti�ened plates.Ahmadian and Bonakdar [18] developed a cylindricalsuper element to performed structural analysis of alaminated hollow cylinder. Taghvaeipour et al. [19]presented the new cylindrical element formulation forthe vibration analysis of FGM hollow cylinders. Theyhave shown that the super element is a capablecylinder, is accurate enough, and is much less timeconsuming.

In this paper, stress analysis of a FGM cylinderand piston under combustion pressure, consideringthermal stresses, is performed. SEM is implemented toanalyze the mechanical behavior of both FG cylindersand pistons in a transient state. Materials are assumedto vary in the radial direction of the cylinder and thelongitudinal direction of the piston. To validate theresults, a simple case of uniform material distribution,metal and ceramic results of the super element, iscompared with ABAQUS software modeling and verygood results were obtained. The distribution of thematerial is based on the power low method, and byapplying the super element, mechanical analysis of theFGM cylinder is evaluated.

2. Formulation and modeling

Governing equations for the cylinder and piston are de-veloped using an energy approach, and, consequently,an appropriated code is developed using MATLABsoftware.

The shape function for the SE before implement-ing this element into the equation of motion should bede�ned. A super element is shown in Figure 1 with 16nodes. In this �gure, r, � and z are radial, tangentialand axial coordinates, respectively.

Considering global and local coordinate systems,the relation between these coordinates may be ex-pressed as follows:

R. Pourhamid et al./Scientia Iranica, Transactions B: Mechanical Engineering 22 (2015) 493{503 495

Figure 1. The super element with 16 nodes.

� =2r � (ri + ro)

ro � ri ; (1)

� =�� 1; (2)

� =2zl; (3)

Ni(�; �; �) = ~fi(�; �; �); (4)

where L is the super element length, and ri and ro areinner and outer radii of the geometry.

The speci�c shape functions for this element aregiven in the Appendix.

2.1. Stress analysisThe relation between the strain and the displacementcould be written as follows [18]:

f"g6�1 = [d]6�3fug3�1; (5)

where:

f"g =�"rr "�� "zz r� rz �z

T ; (6)

fug =�ur u� uz

T : (7)

And, since the strains can be related to the componentsof the displacement vector as [18]:

"rr =@ur@r

; "�� =urr

+1r@u�@�

;

"zz =@uz@z

; (8)

r� =1r@ur@�

+@u�@r� u�

r;

rz =@ur@z

+@uzr;

�z =@u�@z

+1r@uz@�

: (9)

The d matrix is achieved as:

[d] =

2666666666664

@@r 0 01r

1r@@� 0

0 0 @@z

1r@@�

@@r � 1

r 0@@z 0 @

@r

0 @@z

1r@@�

3777777777775: (10)

The strain energy associated with this system can bewritten as [20]:

� =Zvf"gT f�gdv �

ZsfugT f~pg ds; (11)

where:

f~pg =�

~pr ~p� ~pzT ; (12)

f�g = f�rr��zz�r��rz��zgT ; (13)

where f~pg is the combustion pressure and f�g is thestress note.

Due to the principle of minimum total potentialenergy, the di�erential state of this energy shouldbe zero (�� = 0). Hence, considering Eq. (5) andthe Kantorovich approximation for the displacement(fug3�1 = [ ~N ]3�48fu(e)g48�1), Eq. (11) could bereduced to the following equation:Z

v[B]T f�gdv �

Zs

h~NiT f~pg ds = 0; (14)

where:

[B]6�48 = [d]6�3

h~Ni

3�48; (15)

h~Ni

=

24N1 0 0 � � � N16 0 00 N1 0 � � � 0 N16 00 0 N1 � � � 0 0 N16

35 ;(16)

[B] =

26666666664

N1;r 0 0 � � �N1r

N1;�r 0 � � �

0 0 N1;z � � �N1;�r N1;r � N1

r 0 � � �N1;z 0 N1;r � � �

0 N1;zN1;�r � � �

N16;r 0 0N16r

N16;�r 0

0 0 N16;zN16;�r N16;r � N16

r 0

N16;z 0 N16;r

0 N16;zN16;�r

3777777777775: (17)


According to Hook's law, the relation between stressand strain could be considered as [18]:

f�g6�1 = [D]6�6(f"g � f"T g)6�1; (18)

where:

f"T g =��T�T �T�T �T�T 0 0 0

T ; (19)

[D] =E

(1 + �)(1� 2�)266666641� � � � 0 0 0� 1� � � 0 0 0� � 1� � 0 0 00 0 0 1�2�

2 0 00 0 0 0 1�2�

2 00 0 0 0 0 1�2�

2

37777775 ; (20)

in which �T , �T , E and v are the thermal expansioncoe�cient, temperature di�erence, elastic modulus andPoisson ratio, respectively.

Thus, Eq. (14) could be represented as follows:Zv[B]T [D]

�[B]nu(e)

o�f"T g� dv�Zs[N ]T f~pg ds=0:

(21)

Then, the governing equation could be simpli�ed asfollows:h

~k(e)in

u(e)o

=n

~f (e)o; (22)h

~k(e)i

=Zv[B]T [D][B]dv; (23)h

~f (e)i

=Zv[B]T [D] f"T g dv +

Zs

h~NiT f~pg ds: (24)

Boundary conditions for the cylinder could be men-tioned as follows:

uz = o at z = �L=2;ur = o at z = �L=2; r = ro;

�rr = �p at r = ri;

�rr = 0 at r = ro; (25)

where p is the combustion pressure. Here, it is assumedthat the top and bottom areas of the cylinder are in a�xed condition (in the direction of z) in connectionwith the cylinder head and the oil pan in the engine.To prevent the cylinder motion in the direction of thecombustion pressure, the motion of the outer radiusin the top and bottom of the cylinder is assumed tobe zero. In addition, it is assumed that there is nocontraction between the cylinder and piston.

Figure 2. The combustion pressure and the heat uxversus the time.

Boundary conditions for the piston are assumedto be:

uz = o at z = �L=2;�zz = �p at z = +L=2;

�rr = 0 at z = �L=2: (26)

The force from the connecting rod to the piston isneglected here. Instead, it is assumed that the bottomarea of the piston is in a �xed condition (in the directionof z).

2.2. Load conditionsThe combustion pressure and heat generation due tocombustion in the cylinder are shown in Figure 2.In this �gure, the combustion pressure and heatgeneration versus time, for a complete crank shaftcycle (720 degrees), at an engine speed of 3000 rpm,is plotted. The maximum pressure is considered to be3161.3 kPa, which occurred after about 0.02 sec of thecombustion process.

Temperature distribution within the cylinder, andthickness is given by Figure 3(a), while the tempera-ture distribution in the piston length is presented byFigure 3(b). The temperature distribution is evaluatedbased on the heat ux of the engine [22].

2.3. Geometry and propertiesThe geometric dimensions of the cylinder and pistonfor the engine are as follows:

� Piston radius: 71 mm;

� Piston length: 27.9 mm;

� Cylinder inner radius: 71 mm;

� Cylinder outer radius: 81 mm;

� Cylinder length: 83.6 mm.


Figure 3. Temperature distribution for (a) FG cylinder,and (b) FG piston.

The material structure for the cylinder is assumed tostart as ceramic at the inner surface and graduallychanges to metal at the outer surface. For the piston,it starts as ceramic at the top surface in contact withcombustion and changes gradually to metal at thebottom surface.

The material pro�le for the cylinder and pistonis assumed to have a power low distribution startingfrom ceramic and ending with metal; inside-out for thecylinder (Eq. (27)) and top to bottom for the piston(Eq. (28)):

vm(r) =�r � riro � ri

�nc; vc(r) = 1� vm(r); (27)

vc(z) =�

2z � L2L

�np; vm(z) = 1� vc(z); (28)

where Vm and Vc are metal and ceramic volumefractions, respectively. The mechanical and physicalproperties of the material (P ) for the cylinder and

Figure 4. Distribution of the (a) metal volume fractionversus the radius of the cylinder, and (b) ceramic volumefraction versus the length of the piston.

piston may be written as follows:

p(T; r) = pm(T )vm(r) + pc(T )vc(r); (29)

p(T; z) = pm(T )vm(z) + pc(T )vc(z); (30)

in which Pm and Pc are metal and ceramic properties,respectively. Figure 4(a) and (b) represent metalvolume fraction versus the radius of the cylinder andceramic volume fraction versus the length of the piston,respectively.

The mechanical and physical properties of themetal (Ti-6Al-4V) and ceramic (Si3N4) are presentedin Table 1. It should be mentioned that titanium alloy(Ti-6Al-4V) is of great interest to car manufacturers inengine design.

2.4. Validation processTo validate the results obtained by the super element,conventional �nite element software is used for sim-ple cases of a metal-rich state and for a ceramic-rich condition. For the axisymmetric cylinder and


Table 1. Material properties for the studied FGM.

Characteristics Metal(Ti-6Al-4V)

Ceramic(Si3N4)

�� kg

m2

�2370 4429

cp�

Jkg�K

�625.30 555.11

k� W

m�K�

13.72 1.21

E (GPa) 122.56 348.43

v (-) 0.29 0.24

�T� 1�K

�7:58� 10�6 5:87� 10�6

Figure 5. FE models of (a) the piston and (b) thecylinder in the ABAQUS software.

piston, the minimum number of elements and nodesfor the convergence are chosen to be 2100 �nite el-ements, with 2210 nodes for the cylinder and 7952�nite elements, with 8136 nodes for the piston (Fig-ure 5).

3. Results and discussion

3.1. Convergency for cylinderConvergence of the results for a metal-rich (n = 0)cylinder using a super element is presented in Figure 6.As the number of super elements increases beyond 20,the stress distribution variation is minimal.

Figure 6. Distribution of the longitudinal stress in themetal-rich cylinder for di�erent numbers of super elements.

3.2. Validation for cylinderStress distribution from the inside-out, in an axialdirection, for metal-rich and ceramic-rich, based ona super element with 20 elements and conventional�nite element analysis at the top of the cylinder, arepresented in Figure 7. Results for the two methodsare the same with maximum di�erence of 2% whilethe computational time with super element is muchless than the conventional �nite element. Stress dis-tribution in radial and longitudinal are presented inFigure 8(a) and (b), respectively.

Distribution of radial stress (in the mid-length ofthe cylinder) is uniform and changes gradually throughthe radius. The longitudinal stress is compressiveinside the cylinder and changes to tensile in the radialdirection. In addition, the value of the radial stressis lower than the value of the longitudinal stress, farfrom the boundary. The stress value in ceramic-richmaterials is higher than that of metal-rich materials.This is due to the higher value of modulus elasticity(Table 1).

3.3. Convergency for pistonThe radial stress distribution of the metal-rich piston(n = 0) is depicted in Figure 9. As the number ofsuper elements along the length of the piston used inthe model increases, the radial stress gradient increases.Based on the convergency test, it is found that 40 superelements (with 328 nodes) in the length direction of thepiston have enough accuracy for correct radial stressdistribution.

3.4. Validation for pistonStress distribution from the top to bottom and ina radial direction with metal-rich and ceramic-richmaterials, based on a super element with 40 elementsand with conventional �nite element analysis of thepiston, are presented in Figure 10. Results for the twomethods are the same, with a maximum di�erence of


Figure 7. Distribution of (a) the radial stress and, (b)the longitudinal stress in the cylinder for the validation.

2%, while computational time with the super elementis much less than with the conventional �nite element.The radial and longitudinal stress distribution for ametal-rich piston is shown in Figure 11. As illustratedin these �gures, the value of the longitudinal stressis lower than that of the radial stress, far from theboundary. The longitudinal stress is approximatelyuniform throughout the length of the piston. The radialstress is compressive on the top area of the piston,where the maximum temperature occurs, and then itbecomes tensile in other regions.

3.5. SE resultsDistribution of radial stress and longitudinal stress fordi�erent value powers of n in the FG cylinder arepresented in Figure 12.

The radial stress is smaller in comparison withlongitudinal stress, and the value of this stress issmaller than the combustion pressure. From the inner-out side of the cylinder, the radial stress increasesin a compressive state and then decreases to zero on

Figure 8. Distribution of (a) the radial stress and, (b)the longitudinal stress in the metal-rich cylinder obtainedby the ABAQUS software.

Figure 9. Distribution of the radial stress in themetal-rich piston for di�erent numbers of super elements.

the outer radius of the cylinder, where there is noexternal pressure. As nC increases, i.e. changing frompure metal to pure ceramic, the absolute value of theradial stress decreases for all values of cylinder radius.Changing the material of the cylinder from pure metalto pure ceramic leads to a decrease in radial stress, aswell as an increase in longitudinal stress. However, asnC increases, the longitudinal stress increases sharply,from 500 MPa to 1000 MPa, at the inner surface of the


ceramic cylinder. By increasing nC , a larger number ofregions become ceramic, with higher elastic modulus.It is inferred that the e�ect of material distributionis more tangible on longitudinal stress than on radialstress.

Stress distribution versus length of piston fordi�erent values of nP is demonstrated in Figure 13.

Figure 10. Distribution of (a) the radial stress and, (b)the longitudinal stress in piston for the validation.

The radial stress decreases by the enhancement of nP .Higher amounts of nP in the FG piston mean thatthe material will become softer (almost as a metal-rich material). For example, when nP increases, radialstress increases sharply, from 600 MPa to 1100 MPa,on top of the piston. It is also observed that the

Figure 12. Distribution of (a) the radial stress and, (b)the longitudinal stress in the FG cylinder for di�erentvalues of n by using the SEM.

Figure 11. Distribution of (a) the radial stress and, (b) the longitudinal stress in the metal-rich piston obtained by theABAQUS software.


Figure 13. Radial stress distribution in the FG piston fordi�erent values of nP by using the SEM.

e�ect of material distribution on radial stress is verysigni�cant.

4. Conclusion

In this paper, stress analysis of a FGM cylinder and pis-ton of a combustion engine during the transient stageis performed. E�ects of pressure and thermal stressesare included in the problem. A newly developed hollowcylindrical super element with 16 nodes is used for theanalysis. The obtained results for simple cases, withuniform metal or ceramic materials, nicely matchesthe results obtained by the conventional �nite elementABAQUS package. In the modeling with a super ele-ment, for similar �ndings, approximately �ve percentof the number of elements in the conventional �niteelement cylindrical type element is needed, while thetime consumed is much less than in the conventionalelement.

Distribution of the material is assumed to be inthe form of the power law. Considering the nonlineartemperature distribution in the cylinder, the thermalstresses at the internal surface of the cylinder are alsononlinear, which will be added to the pressure of thecylinder due to combustion. Findings also indicate thatby changing the power of the material distribution,stress distribution can also be optimized.

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Appendix

Shape functions [19]:

N1(�; �; �) =18

(cos2 �� cos��)(1 + �)(1 + �);

N2(�; �; �) =18

(cos2 �� cos��)(1� �)(1 + �);

N3(�; �; �) =18

(sin2 �� sin��)(1 + �)(1 + �);

N4(�; �; �) =18

(sin2 �� sin��)(1� �)(1 + �);

N5(�; �; �) =18

(cos2 ��+ cos��)(1 + �)(1 + �);

N6(�; �; �) =18

(cos2 ��+ cos��)(1� �)(1 + �);

N7(�; �; �) =18

(sin2 ��+ sin��)(1 + �)(1 + �);

N8(�; �; �) =18

(sin2 ��+ sin��)(1� �)(1 + �);

N9(�; �; �) =18

(cos2 �� cos��)(1 + �)(1� �);

N10(�; �; �) =18

(cos2 �� cos��)(1� �)(1� �);

N11(�; �; �) =18

(sin2 �� sin��)(1 + �)(1� �);

N12(�; �; �) =18

(sin2 �� sin��)(1� �)(1� �);

N13(�; �; �) =18

(cos2 ��+ cos��)(1 + �)(1� �);

N14(�; �; �) =18

(cos2 ��+ cos��)(1� �)(1� �);

N15(�; �; �) =18

(sin2 ��+ sin��)(1 + �)(1� �);

N16(�; �; �) =18

(sin2 ��+ sin��)(1� �)(1� �):

Biographies

Reza Pourhamid received BS and MS degrees inMechanical Engineering from Guilan University, Rasht,Iran, and Mazandaran University, Babol, Iran, in 2001and 2005, respectively. He is currently a PhD degreecandidate in the Science and Research Branch of theIslamic Azad University, Tehran, Iran. His researchinterests include heat transfer, combustion and thermo-elasticity.

Mohammad Taghi Ahmadian received his BS andMS degrees in Physics, in 1972, from Shiraz Univer-sity, in Iran. He also completed the requirementsfor BS and MS degrees in Mechanical Engineeringreceiving his MS degree in 1980 from the Universityof Kansas in Lawrence, USA. He also obtained PhDdegrees in Physics and Mechanical Engineering, in1980 and 1986, respectively, from the same institution.He is currently Professor in the School of Mechan-ical Engineering at Sharif University of Technology,Tehran, Iran, where he is also Head of the Bio-Engineering Center. His research interests includemicro and nano mechanics, as well as bioengineer-ing.

Hosein Mahdavy-Moghaddam received his PhDdegree in Mechanical Engineering in 2000 from theUniversity of Utah (Salt Lake City, Utah, USA), wherehe also taught under graduate courses. He is currentlyAssistant Professor in the Mechanical and AerospaceEngineering Department of the Science and Research


Branch of the Islamic Azad University, Tehran, Iran.His research interests are heat transfer and combus-tion.

Ali Reza Mohamadzadeh received his PhD degreein Mechanical Engineering from Michigan University,

USA, and is currently Associate Professor and Headof the Mechanical Engineering Department in theScience and Research Branch of the Islamic AzadUniversity, Tehran, Iran. His research interests in-clude thermo-elasticity, thermo-magno-elasticity andnonlinear-dynamics and vibrations.

Mechanical analysis of a functionally graded cylinder...

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