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CURRICULUM VITAE
PERSONAL DETAIL
Name Dr. Ramesh Kumar Lalwani
Designation Senior Research Fellow
Department Department of MechanicalEngineering
Faculty Faculty of Engineering
Tel. No. (Office) 60379676815
Fax No. +60379675317
E-mail Address drlalwani@um.edu.my
Homepage URL
mechanical.eng.um.edu.my/ramesh...
ResearcherID Link C-8659-2009 112508...
Address(Office) Department of MechanicalEngineering, Faculty of EngineeringUniversity of Malaya, 50603 KualaLumpur, MALAYSIA
ACADEMIC QUALIFICATION(Qualification), (Institution).
Ph.D. (Mechanical Engineering)(IIT Kharagpur), INDIAN INSTITUTE OF TECHNICAL,KHARAGPUR
M. tech, INDIAN INSTITUTE OF TECHNICAL, KHARAGPUR
B. Sc. [Engineering], VIKRAM UNIVERSITY, UJJAIN, INDIA
PROFESSIONAL AFFILIATION/MEMBERSHIP(Organisation), (Role), (Year), (Level).
NFPA, Member, 2000, (National)
INSTITUTE OF ENGINEERIS [INDIA], Honorary Fellow, 1995, (National)
THE INDIAN SOCIETY FOR TECHNICAL EDUCATION, Life Member, 1993,(National)
CAREER HISTORY(Post), (Organisation), (Period).
Adiministration, Teaching, Research And Consultancy, SKYLINE INSTITUTE OF
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ENGINEERING AND TECHNOLOGY, 22/05/2008 until 05/06/2009
AREAS OF EXPERTISE(Area).
Noise And Vibrations (Design, Noise and Vibrations)
Acoustic Engineering (Noise, vibrations, dynamic stress analysis, )
RECENT SELECTED PUBLICATIONS(Publication).
Academic Journals
2010
reza afshar; M. Bayat, Dr.; R. K Lalwani, Prof.; Y.H. Yau, Associate Prof. (ISI-CitedPublication)
2009
Proceeding IUTAM Symposium on Emerging Trends in Rotor Dynamics (IUROTOR-2009), March 23-26, 2009, Indian Institute of Technology Delhi, India. (ISI/SCOPUSCited Publication)
Proceeding IUTAM Symposium on Emerging Trends in Rotor Dynamics (IUROTOR-
2009), March 23-26, 2009, Indian Institute of Technology Delhi, India. (ISI/SCOPUSCited Publication)
1982
LALWANI, R. K. (ISI-Cited Publication)
1971
Lalwani, R. J. (ISI-Cited Publication )
AREAS OF RESEARCH(Project title), (Role), (From)-(Until), (Source), (Level).
LARGE DEFLECTION, THERMO-MECHANICAL ANALYSIS AND SIMULATIONOF FUNCTIONALYY GRADED PIEZOELECTRIC PIPES BASED ON SHEARDEFORMATION THEORY, Principal Investigator(PI), 2010-2011
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CONSULTATION PROJECT/CONSULTANCY(Project title), (Role), (From)-(Until), (Organisation).
Dynamic Analysis of Water Turbine, Project Leader, Skyline Institute of Engineering andTechnology
AWARDS AND RECOGNITIONS(Name of Award), (Awarding Institution), (Year Awarded), (Level).
Best Speaker Award, MAULANA AZAD NATIONAL INSTITUTE OF TECHNOLOGY,1982
HSIEH-CHIH ASSOCIATION AWARD, TATUNG INSTITUTE [NOW UNIVERSITY]OF TECHNOLOGY, 1989
PRESENTATIONS(Title), (Event), (Date Presented), (Organiser), (Level).
Presenter
Role of Research and Morality in Quality Learning in Higher Education., InternationalCONFERENCE ON TEACHING AND LEARNING IN HIGHER EDUCATION 2009,13 Nov 2009 to 13 Nov 2009, ICTLHE09
Role of IT in Educational Institutes, Role of IT in Educational Institutes , March 17-18,2001, Fifth Annual Convention of ISTE (M.P.), National Seminar on Impact of
Information Technology in Technical Education, Indore, India, 17 Mar 2001 to 17 Mar2001, Indian Society for Technical Education, Bhopal., (National)
Problems and Issues in Realizing Sustainable Development In Higher Education ThroughSoft and Hard Skills, THE 3RD INTERNATIONAL CONFERENCE OF UNESCOCHAIR HIGHER EDUCATION FOR SUSTAINABLE DEVELOPMENT (HESD):TRANSFORMING HIGHER EDUCATION FOR A SUSTAINABLE SOCIETY, 20-22
NOVEMBER 2009, USM, PENANG, MALAYSIA., 21 Nov 2009 to 21 Nov 2009,USM, PENANG, MALAYSIA., (International)
(b)Rotor Fault Detection in Machines: Methods and Techniques, IUTAM Symposium onEmerging Trends in Rotor Dynamics (IUROTOR-2009), March 23-26, 2009, IndianInstitute of Technology Delhi, India., 25 Mar 2009 to 25 Mar 2009, Indian Institute ofTechnology Delhi, India., (International)
On Condition Machine Health Monitoring Against Corrosion., World CORCON 2009"International Conference & Expo on Corrosion during 29th Sept. - 1st Oct 2009 atMumbai, 30 Sep 2009 to 30 Sep 2009, World CORCON 2009, (International)
SUPERVISION
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Post Graduate Level(Name of Degree), (Name of Candidates), (Title of Thesis), (Academic Session), (Status)
Master Degree, Lee, FINITE ELEMENT ANALYSIS OF SPUR GEAR TOOTH,1989/1990, Completed
Master Degree, Sajjad Zarghan, VIBRATIONAL ANALYSIS OF TAPPEREDCRACKED SHAFT, 2009/2010, Ongoing
TEACHING(Course Title), (Academic Session), (No of Student), (No of Contact Hours).
First Degree
COMPUTATIONAL MATERIAL ENGINEERING, 2009/2010(1), 21, 72
MECHANICS OF APPLIED MATERIALS, 2009/2010, 61, 72
NON-FERROUS ALLOY, 2009/2010(2), 60, 52
FLUID DYNAMICS, 2007/2008(2), 100, 98
EVALUATION ACTIVITIES(Description), (Evaluation Activity).
Under graduate, Post gradute and Ph.D. Theses and viva presentation, Thesis
Effect of Nickel and Cobalt Nanoparticle additions to Sn-Ag-Cu Solder, Thesis
BIOGRAPHY
Prof. Ramesh Kumar Lalwani received his PhD in 1972 from IndianInstitute of Technology, Kharagpur based on his doctoral thesis titled"Noise and Vibrations generated by Ball Bearings". For the next two years,he led the research group on "Elastic Wave Propagation in rods" at theinstitute fur Mekanik at Technical University Hannover, Germany. In 1975he participated at International Centre for Mechanical Sciences, Udine,Italy. In 1976 he was appointed as expatriate visiting faculty at Universityof Benin, Benin City. In 1981 he joined as Head of Agriculture EngineeringDepartment at the Federal University of Technology, Makurdi. He was thenappointed as the dean of the Faculty of Engineering at University of Jos, Josin 1985. In 1986 he was selected to join Tatung University of Technology,
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Taipei, Taiwan where as Professor and consultant to Tatung Company hecontributed in many research projects and publication along with teachingand guiding post graduate students. In 1989 was awarded with a citationwhich states " Professor Ramesh Kumar Jamnadas Lalwani of TatungInstitute of Technology has devoted himself to scholastic research and
published many outstanding papers in different academic journals. We arehereby honored, in recognition of his great contributions to Science andTechnology, to present him this Hsieh-Chih Association Award". In 1991he joined Gajra Bevel Gears, Dewas as their ISO-9001 consultant. In 1992he was appointed as the Principal of A. C. College of Engineering,Sangamner. Since then he has been working in many Indian Institutes ofTechnical education. He has published 3 books, many memo-graphs, andresearch papers. He is a life member of Indian Society for TechnicalEducation and Fellow of the Institute of Engineers [India}, Member ofInternational NFPA, USA.Since 2009 he is professor of Mechanical Engineering in top Malaysian and
among the top 180 universities of the World the University of Malaya.
Elastic behavior of glass-like functionallygraded infinite hollow cylinder underhydrostatic loads using finite element method
R. Afshar, a,
, M. Bayata,
, R.K. Lalwania
and Y.H. Yaua
a Mechanical Engineering Department, University of Malaya, 50603 Kuala Lumpur, Malaysia
Received 16 March 2010;
accepted 19 July 2010.
Available online 24 July 2010.
Abstract
A glass-like (viscoelastic) functionally graded cylinder is studied by using finite element methodto investigate the mechanical responses. A subroutine is developed by using ANSYS parametric
design language (APDL) to simulate two nonlinearities, which are the variation of material
properties with respect to time and position. The cylinder is made of two different viscoelastic
materials, namely, pure material one at inner and pure material two at outer surfaces. The
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material properties are assumed to be presented by simple power law distribution and moreover,
bulk and shear moduli are varying with respect to time using the kernel functions depicted
regarding Prony series. It is shown that the hoop stresses take the same values at the mean radius
(middle of the thickness) for different values of time and grading index. It is found that the radial
stress decreases to certain values for specific grading index and then by increasing the grading
index it increases to maximum value that related to pure material cylinder. It is shown that unlike
the zero axial stress in pure material cylinders, it varies along the thickness from minimum to
maximum at inner and outer surfaces, respectively. It is concluded that the viscoelastic
functionally graded (VFG) materials play an important role in steady and transient response of
hollow cylinder under hydrostatic load.
Keywords: A: Glass-like functionally graded materials; E: Mechanical response; F: Elasticbehavior
Article Outline
1.Introduction
2.Gradation relations
2.1. Viscoelastic material properties2.1.1. Constitutive equations
2.2. Functionally graded materials (FGMs)
3.Numerical method
3.1. Material properties
3.2. Element types
4.
Boundary conditions5.Numerical results for the case study and discussion5.1. Results and discussion
6.Conclusions
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References
1. Introduction
A material with elastic and viscous parts is defined as viscoelastic material. Generally,
viscous property changes with respect to time, whereas the elastic deformation occursinstantaneously due to applied load. The brittle materials such as glass or glass-like
materials by having different properties due to heating and cooling process can be
modeled as viscoelastic materials [1], [2] and [3]. The brittle materials with high values
of modulus of elasticity can be applied to reinforce the pure metal structures. This
arrangement of two different materials introduces advance material named functionally
graded materials (FGMs). FGMs are combination of at least two materials which vary
smoothly as a function of location along certain dimension(s) of the structure by
considering variation of volume fraction of components [4], [5], [6], [7] and [8].
Application of FGMs can be seen in pipes in oil and gas industry, under water
equipments, aerospace instruments, storage cylinders, hollow rotating shafts and
winding of composite pressure vessels and so on. Because of increasing application of
FGMs and viscoelastic materials, new methodologies need to be developed to
characterize, analyze and design structural components made of these materials.
A number of investigations dealing with mechanical loads like hydrostatic, dynamic as
well as thermal loads have been published in the scientific literature [9], [10], [11] and[12]. In recent years, Vinogradov and Milton [9] studied the creep of a composite
consist of two linear viscoelastic materials and subjected to a constant hydrostatic or
anti-plane loading. Aydlner and stndag [10] investigated the residual stresses in a
bulk metallic glass cylinder. They induced thermal tempering to the model and
analyzed the stress generation. Lee [11] examined the thermomechanical response of a
viscoelastic thin-walled cylinder under instant internal pressure while uniform
temperature increased gradually by using a time-domain boundary element analysis.
Seoudi et al. [12] solved the two-dimensional elastic wave equations of the viscoelastic
cylinders to investigate the periodic deformations of the cylinders. Golden and Graham
[13] studied a dynamic response of viscoelastic rolling cylinder using the non-inertial
approximation. Different types of advanced materials are used for cylindrical shapes
for different purpose, for example biological tissues with viscoelastic behavior in
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biomechanics field as well as functionally graded (FG) cylinder used in oil and gas
transportation.
Many studies have been done FG cylinders due to internal pressure [14], [15], [16],
[17], [18], [19] and [20]. Sepiani et al. [14] studied the cylinders with combination of
FG and pure material shell as inner and outer, respectively, to obtain the free vibration
and buckling results while cylinders subjected to fatigue loads. Theoretical
formulations were obtained based on First-order shear deformation theory and their
results were verified with numerical method. Hosseini [15] applied Galerkin Finite
Element (FE) to investigate the coupled thermoelastic behavior of FG thick hollow
cylinder. The dynamic behavior as well as thermomechanical response were obtained.
Later, Shahabian and Hosseini [16] applied the same procedure such as [15] as well as
Monte Carlo simulation to study a dynamic behavior of a FG thick hollow cylindersubjected to shock loading. A non-linear power function of radius was assumed for
variation of the mechanical properties along the thickness of structure. Batra and
Laccarino [17] found elasticity solutions for displacement field in a long FG cylinder
subjected to hydrostatic pressure. The material properties were assumed an isotropic
and incompressible second-order elastic material with modulus varying only in the
radial direction. Huang and Han [18] considered a long FG cylindrical shell to describe
post-buckling behavior and also the non-linear buckling subjected to axial compression
and lateral loads. They employed Ritz energy method to present the non-linear large
deflection. The numerical results showed various effects of the inhomogeneous
parameter, dimensional parameters and external thermal environments. In another
study, Batra and Bahrami [19] investigated the radial deformation of a FG hollow
cylinder under internal pressure. MooneyRivlin material model was considered to
describe the material properties. Shen [20] studied a FG cylindrical shell under torsion
loading to analyze post-buckling behavior in thermal environment. A higher order
shear deformation theory was used for the governing equations. The results revealed
that distribution of the volume fraction of FGMs has a noteworthy effect on the
buckling load and post-buckling behavior of FGM cylindrical shells under torsion.
However, for some specific application such as in biomechanics and aerospace where
high toughness, light weight and durability becomes crucial, the components need to be
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fabricated using special material such as a viscoelastic functionally graded materials
(VFGMs).
Many studies on VFGMs have been reported [21], [22], [23], [24], [25], [26], [27],
[28] and [29]. Khan and Muliana [21] presented a micromechanical-structural
framework to analyze VFGMs under thermomechanical loads. They compared the
experimental and numerical result. Panda and Ray [22] investigated a FG laminated
composite plates integrated with a patch of active constrained viscoelastic layer
damping treatment to obtain the non-linear dynamic response. Pan et al. [23] studied
the fracture analysis of a viscoelastic functionally graded (VFG) strip. Parameters such
as material grading index, crack length and spacing as well as the loading condition on
the crack tip field intensity factor were investigated. A micromechanical model was
introduced by Muliana [24] for predicting effective thermal behaviors of VFGMs.Experimental data as well as analytical solutions in the literatures were used for
verification of their work. Khazanovich [25] showed the compatibility of elastic
viscoelastic correspondence principle for non-homogeneous materials with separable
relaxation modulus. Gilhooley et al. [26] used the meshes local PetrovGalerkin
(MLPG) procedure to simulate a two-dimensional static and dynamic deformation of
VFGMs. Bhangale and Ganesan [27] used the finite element method (FEM) to study
the bucking and vibration behavior of a VFG beam in thermal environment. Hilton [28]
modeled FGMs structure as non-homogenous material and investigated viscoelastic
and elastic behavior of the structure. In another study, Sladek et al. [29] analyzed two-
dimensional anisotropic and linear VFG solids by using the MLPG approach.
Moreover, he verified the accuracy of the proposed method.
It can be noted that the existing literature on viscoelastic and FG cylinders as well as
VFGMs structures, to the best of authors knowledge, a very limited and little work has
been done that analyzed VFG long cylinder under hydrostatic load. This very fact
motivates the investigation of the present study. This study attempts to consider an
axisymmetric model of long thick-walled cylinder of inner radius Ri and outer radius
Ro subjected to internal hydrostatic pressure,Pi (Fig. 1). The viscoelastic material
properties of the FG cylinder are varying along the thickness based on a simple power
law distribution in terms of the volume fractions of the constituents (detail A in Fig. 1
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illustrates the number of layers along the thickness of the cylinder). FEM is used in
ANSYS environment to obtain radial deformation and stress as well as hoop stress for
various materials grading index (MGI). A subroutine is developed using APDL. The
cylinder is made of two different materials at the outer and inner surfaces to investigate
the effect of gradual variation of material properties. Viscoelastic parameters, bulk and
shear moduli, are varying with respect to time using the kernel functions depicted
regarding Prony series while they are function of radial position by using power law
distribution.
Full-size image (31K)
Fig. 1. Configuration of a long VFG cylinder.
View Within Article
2. Gradation relations
2.1. Viscoelastic material properties
2.1.1. Constitutive equations
Based on infinitesimal theory, in the range of fast and slow applied load, elastic
response can be calculated by the Caushy stress () as:
(1) where G(t),K(t), e, , t, and Iare shear relaxation kernel function, bulk relaxation kernel function, deviatoric part of
the strain, volumetric part of the strain, current time, past time and unit tensor,
respectively.
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For the element used in this analysis (PLANE183) the kernel functions areKand G
depicted regarding Prony series, which assumes that:
(2) where
K,Ki, , and Gi are bulk elastic, moduli shear relaxation times elastic moduli
for each Prony component and trepresents the current time. The number of Prony
terms for shear is nG and for volumetric behavior is nK. The relaxation behavior of the
deviatoric and volumetric portions of the stress is different. It is worth mentioning that
the value ofnG, nK, and can be different.
The relative moduli are introduced as:
(3) where and
.
The kernel functions can be equivalently expressed as:
(4) Here, G0, G andK0,K, are the shear
and bulk moduli at the fast and slow load limit, respectively.
2.2. Functionally graded materials (FGMs)
In this study, the property variation, M, of the material in the VFG cylinder along the
radial direction is assumed based on power law distribution of the following form [4]:
(5) Here Mi and Mo are the
material properties of inner and outer surfaces of hollow cylinder, respectively, Mo is
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the material property of pure material one and Mi is the material property of pure
material two; n 0 is the gradation index.
3. Numerical method
The basis of simulation has been carried out by using ANSYS finite element package.
The following section describes in detail the steps involved in using the software for
analysis of VFG cylinder.
3.1. Material properties
The mechanical properties needed are modulus of elasticity, Eand Possions ratio, .
Two Proney series in representation of viscoelastic behavior are specified using TB
and TBDATA commands in ANSYS.
3.2. Element types
An axisymmetric 2-D eight-noded PLANE183 element is used (Fig. 2). Two types of
PLANE183 with eight or six nodes with quadratic displacement behavior can be used.
And also, the option of plane stress or plane strain can be applied for the axisymmetric
models in this element. In the axisymmetric case, Yaxis is the axis of symmetry [1].
Full-size image (18K)
Fig. 2. PLANE183 element geometry [1].
View Within Article
For the purpose of simulation, the long cylinder is divided into a number of divisions in
radial direction. Each of the divisions has its own material properties in accordance
with power law distribution in the radial direction as it is mentioned in Eq. (5). The
material properties of the FG cylinder are evaluated at the element centroid. ANSYS
Parametric Design Language (APDL) is used.
Fig. 3 shows the flowchart of simulation of VFG cylinder.
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Full-size image (67K)
Fig. 3. Flowchart of computational scheme.
View Within Article
4. Boundary conditions
The VFG cylinder is subjected to inner pressure ofPi as shown in Fig. 1. The
following traction boundary conditions on the hollow cylinder must be satisfied:
where Ua and r are axial displacement and radial stress,
respectively.
5. Numerical results for the case study and discussionFor numerical illustration of the non-linear elastic solution of this study, a hollow
viscoelastic cylinder with Ri/Ro = 0.5 and thickness of 2 (50 mm), subjected to
internal pressure of 1 Psi (6895 Pa) is considered. The cylinder is long in the out-of-
plane,Z, direction. The axisymmetric model of the cylinder with applied loads and
boundary conditions is depicted in Fig. 4.
Full-size image (40K)
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Fig. 4. Axisymmetric model of the cylinder.
View Within Article
In the following section, results are presented by considering simple case of Eq. (1)
where the shear and bulk moduli of the viscoelastic cylinder behave as:
(6-A)
(6-B) In which G0 andK are defined as
G0 = Ei/(1 + 2c),K = Ei/[3(1 + 2c)] for inner surface and G(0) = G0 = Eo/(1 + 2c),
K0 = Eo/[3(1 + 2c)] for outer surface. The elastic moduli of inner and outer surface are
assumed Ei = 105 Psi (689.5 MPa) and Eo = 20 105 Psi (13789.5 MPa), respectively.The Poissons ratio is constant (c = 1/3).
For more specific to adjust parameters in ANSYS by considering the viscoelastic
material behavior as:
(7) In
accordance with Eq. (6-A), the parameters in Eq. (7) take the following values:
The material properties are given in Table.
1.
Table 1.
Material properties used for case study.
Material property E (MPa) G1/G0 K1/K0 G/K
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Material property E (MPa) G1/G0 K1/K0 G/K
Pure material one (inner surface) 689.5 0.3 1 0.0 1.0
Pure material two (outer surface) 13789.5 0.3 0.5 0.5 2.0
Full-size tableView Within Article
In this section, results are presented in non-dimensional form normalizing stresses and
radial displacement by factors Eo andRo, respectively. Moreover, the presented method
and results in the following section may be verified by comparing the numerical results
presented in [Fig. 7], [Fig. 9], [Fig. 10] and [Fig. 12]. These comparing and validating
will be a part of our study numerical results reported in the following section.
Full-size image (19K)
Fig. 7. Variation of non-dimensional hoop stress (h/Eo) of VFG long cylinder for
specific grading index n = 0.8 at different times.
View Within Article
Full-size image (18K)
Fig. 9. Variation of non-dimensional radial displacement (Ur/Ro) in VFG long
cylinder for different grading index at the end of solution time (t= 10 (s)).
View Within Article
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Full-size image (21K)
Fig. 10. Variation of non-dimensional radial stress (r/Eo) for different grading index
in VFG long cylinder at t= 10 (s).
View Within Article
Full-size image (17K)
Fig. 12. Variation of non-dimensional axial stress (a/Eo) for different grading index
in VFG long cylinder at t= 10 (s).
View Within Article
5.1. Results and discussion
Fig. 5 illustrates the non-dimensional radial displacement in VFG long cylinder fordifferent times and grading index n = 0.8.
Full-size image (19K)
Fig. 5. Variation of non-dimensional radial displacement (Ur/Ro) in VFG long
cylinder for different times and grading index n = 0.8.
View Within Article
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It can be seen that the radial displacement decreases along the thickness of the VFG
cylinder from inner towards outer. It is observed that the radial displacement increases
with the increase of the time. It is evident that the slope of decreasing radial
displacement increases with respect to time at the certain position. And also the slope
of radial displacement versus radius close to outer surface tends to zero. This
phenomenon can be explained by interaction between material behavior such as
viscoelastic and FGMs by considering the boundary conditions. As expected the inner
displacements are greater than those at outer surface.
Fig. 6 presents the non-dimensional radial stress in VFG long cylinder for different
times and grading index n = 0.8.
Full-size image (18K)
Fig. 6. Variation of non-dimensional radial stress (r/Eo) in VFG long cylinder for
different times and grading index n = 0.8.
View Within Article
Fig. 6 shows the decrease of radial stress along the thickness of the cylinder from inner
towards outer surface. As expected, the value of radial stress is zero at outer surface
because of non-loading condition. Furthermore, the slope of radial stress decreases
with respect to time at the certain position. It is worth to mention that by increasing the
time the shape of variation radial stress along the thickness changes from concave to
convex.Fig. 7 demonstrates the non-dimensional hoop stress of VFG long cylinder for specific
grading index n = 0.8 in different times. Comparing with the results ofFig. 8 in [30], it
can be said that the hoop stresses for the FG cylinder has the same value at certain
point and the dimensionless hoop stresses increase by increasing the time at outer
surface in contract to behavior at inner surface.
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Full-size image (18K)
Fig. 8. Variation of non-dimensional axial stress (a/Eo) of VFG long cylinder for
specific grading index n = 0.8 in different times.
View Within Article
It is noticed that the hoop stresses decrease with the increase of time at each point
while R/Ro < 0.75. Whereas, the hoop stresses increase with respect to time when
R/Ro > 0.75. The slope of hoop stress decreases from inner towards outer surface. It can
be noted that the value of hoop stress for different times is the same at middle of the
thickness.
Fig. 8 depicts the variation of non-dimensional axial stress in VFG long cylinder for
specific grading index n = 0.8 at different times while the inner pressure is applied.
It can be explained that the behavior of variation of axial stresses is the same as that for
hoop stress as shown in Fig. 7. It is noticed that the axial stress gets the same valuebefore the middle of thickness 0.7 < R < 0.75. This phenomenon can be explained by
interaction between viscoelastic as well as FGMs behavior. It is worth mentioning that
the axial stress is negative close to inner surface while there is tensile stress close to
outer surface. This behavior can be justified by presentence of harder material at outer
surface for reinforcing the cylinder.
Fig. 9 illustrates the non-dimensional radial displacement in VFG long cylinder for
different grading index at the end of solution time (t= 10 (s)).
It can be observed that the radial displacement increases with increase of grading
index, n, from zero to (pure material 1) up to its maximum value forn (pure
material 2). It is noticed that the results for VFG cylinder lie in between pure material
cylinders. It can be seen that the radial displacement decreases along the thickness of
the VFG cylinder from inner towards outer for all values of grading index. It is evident
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that the slope of decreasing radial displacement increases with increase of grading
index at the certain position. And also the variation of radial displacement close to
outer surface tends to zero for all grading indexes. It has been noticed from numerical
simulations that radial displacement in pure viscoelastic cylinder (material 1) is greater
than others. Then, in order to show clearly the effect of different values of grading
index (n) the results for pure viscoelastic cylinder (material 1) has not been drawn in
Fig. 9. This result can be validated with the one reported earlier[31], FG cylinder
under radially symmetry loads.
Fig. 10 shows the non-dimensional radial stress for different grading index in VFG
long cylinder at t= 10 (s)
It can be noted that the radial stresses for pure material are the same while the results
for VFG cylinder are smaller than those for pure material cylinders. It is seen that byincreasing the value of grading index, n, the radial stresses decrease to certain values
for certain grading index, n, then increase to pure material cylinder again. These results
confirm accepted results for free boundary condition at outer, that means, stress is zero
where R = Ro. Yet again, the behavior of variation of radial stresses for different times
(Fig. 6) are the same as those for different grading index (Fig. 10). It is indeed when
one considers the pure viscoelastic cylinder (materials 1 or 2), the values of the radial
stresses take the same value and also greater than those for VFG cylinders. Fig. 3 in
Ref. [31] also validates our solution for FG cylinder since the compressive stresses
through out the thickness reach to zero at outer surface by changing the grading index.
Fig. 11 demonstrates the non-dimensional hoop stress for different grading index in
VFG long cylinder at t= 10 (s).
Full-size image (18K)
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Fig. 11. Variation of non-dimensional hoop stress (h/Eo) for different grading index
in VFG long cylinder at t= 10 (s).
View Within Article
It can be seen that the hoop stress for pure material decreases along the thickness of the
cylinder. Furthermore, hoop stress increases by increasing grading index close to outer
surface. It is worth to mention that maximum hoop stress for pure material cylinders
occurs at inner while it occurs at outer surface for FG cylinders. It is seen that the hoop
stress in FG cylinder are less than those in pure material cylinder close to inner surface
in contrast of its behavior close to outer surface.
Fig. 12 illustrates the non-dimensional axial stress for different grading index in VFG
long cylinder at t= 10 (s).It is seen that the axial stress in pure material cylinder remains constant along the
thickness unlike its behavior for VFG cylinder. Furthermore, close to outer surface,
axial stress increases by increasing grading index. It is worth to mention that maximum
and minimum axial stress in VFG cylinder occurs at outer and inner surfaces,
respectively. The presented results can be verified by comparing the axial stresses for
pure material cylinder with the given formula in chapter 11 and Eq. (11.19) by Boresi
and Schmidt [32] as:
(8) It is interesting to mention that the
results for pure viscoelastic cylinder without thermal load (T) are zero. It can be said
that the general behavior of axial and hoop stress for VFG cylinders are similar to each
other.
6. ConclusionsA stress analysis of viscoelastic functionally graded (VFG) cylinders is implemented
by using finite element method (FEM). The conclusions from the present study for
VFG cylinder can be summarized as follows:
For all time and grading index, the radial displacement takes constant values close to
outer surface when the values of grading indexes (n) are smaller than one. Moreover,
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contrary to the results for radial stress, hoop stress can take the positive values
throughout the thickness in different times. Also, the curve of radial stress is convex
unlike hoop stress at all times. The hoop stresses cross each other around the mean
thickness (R/Ro 0.75) at different times, while, this phenomena occurs before mean
thickness (R/Ro 0.725) for axial stresses. The behavior of axial and hoop stress are
the same unlike the radial stress that is always compressive for different time. Unlike
VFG cylinders, maximum hoop stress for pure cylinders takes place at inner surface.
Furthermore, maximum hoop stress in VFG cylinders happens at outer surface while
there is the minimum hoop stress for pure cylinder.
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