![15
2 Closed Form Solution
for the Vibrational
Analysis of Metal–
Ceramic-Based Porous
Functionally Graded Plate
Yogesh Kumar, Ankit Gupta, and Dheer Singh
Shiv Nadar University
2.1 INTRODUCTION
Functionally graded (FG) materials are the superior tailored hybrid materials in which
the properties of the material vary gradually in the preferred direction. Extensive
literature has been published in the area of FG structures. In this context, Zhang
et al. [1] presented an extensive review based on the buckling and vibrational response
of the functionally graded plate (FGP). Gupta et al. [2–4] analyzed the vibrational
behavior of a porous gradient plate using hybrid higher-order shear deformation
theory (HSDT). Kanuet et al. [5] reviewed on smart functionally graded materials
(FGMs) and analyzed buckling and vibration analysis based on fracture problems.
Kurtaran et al. [6] studied the vibrational response of FGP and showed the effects
of vibrating shape. Thai et al. [7] explored the vibrational and bending analysis of
FGP using simple HSDT. Gupta and Talha [8] have studied and analyzed the effect
of porosity on the structural response of a FGP using HSDT. They have reported
that the fundamental frequency of the porous plate increases as the volume fraction
CONTENTS
2.1 Introduction 15
....................................................................................................
2.1.1 Mathematical Formulation 16
..................................................................
2.1.1.1 Displacement Field 16
...............................................................
2.1.1.2 Energy Equations 18
.................................................................
2.1.1.3 Constitutive Relation 21
............................................................
2.1.1.4 Analytical Solution 23
..............................................................
2.2 Results and Discussion 25
...................................................................................
2.3 Conclusions 31
.....................................................................................................
References. 31
...............................................................................................................](https://image.slidesharecdn.com/92989-250428105930-5e021169/85/Characterization-Testing-Measurement-and-Metrology-1st-Edition-Chander-Prakash-31-320.jpg)
![16 Characterization, Testing, Measurement, and Metrology
index (VFI) increases. Significant changes have been reported in frequency for thin
and thick plates. Shahverdi and Barati [9] have analyzed the free vibrational response
of porous FGP. To investigate the vibrational response of a porous FGP on an elastic
substrate, a nonlocal strain gradient elastic model has been developed. The authors
have reported that the influence of various parameters such as nano-pores, nonlocal
parameters, and gradient index have influence on fundamental frequencies of FGP.
Kumar et al. [10] proposed and carried out the geometrically non-linear analysis
of the FGP using HSDT. Zenkour et al. [11] used the 3D elasticity solution for an
Exponentially functionally graded plate (E-FGP). Birman V. et al. [12] proposed the
research article on the modeling and investigation of FG structures. Reddy et al. [13]
emphasized the comprehensive investigation of FGP. Benveniste et al. [14] suggested
a new methodology for the use of Mori-Tanaka’s principle FGP. Mori T et al. [15]
reported average stresses in the average elastic energy in various materials.
Shariat et al. [16] presented the stability analysis of FGP with geometrical imper-
fection based on Classical plate theory (CPT). While using FSDT, Lanhe [17]
derived the equilibrium and stability equations for a simply supported rectangular
FGP with moderate thickness under thermal loads. Javaheri at al. [18,19] obtained
the equilibrium and stability equation using variational approach method which is
based on the CPT for perfect simply supported FGP. The authors have also derived
the equilibrium equations under thermal loads based on the HSDT and power-law
gradient rule. M Mohammadi et al. [20] developed a method to decouple the stability
equation of moderately thick FGP using levy-type boundary conditions. An efficient
and simply refined plate theory for studying buckling behavior of FGP using CPT
was developed by Thai et al. [21]. Shariat et al. [22,23] introduced the rectangular
FGP with or without geometric defects for mechanical and thermal analyses with the
assumption of non-homogeneous mechanical properties varying through the plate
thickness under the compression, tension, and thermal loading.
In this research work, the algebraic-based HSDT has been considered for the
vibrational analysis of FGP. The power-law distribution and Hamilton’s variational
principle have been used to derive the governing equations. Governing equations
are solved using Navier’s method. The effects of various parameters like VFI and
geometric parameter on the fundamental frequency of FGP have been reported in
subsequent section.
The model of the porous FGP has been shown in Figure 2.1. The material proper-
ties of the porous FGP are shown in Table 2.1.
2.1.1 MatHeMatical FoRMulation
2.1.1.1 Displacement Field
The displacement field, i.e., u i
i for 1
= ,2,3, is the function of x, y, z, and t used in this
work for the functional grade plate, reported by H.T. Thai [7], and expressed as the
following equations:
∂w 4z3
∂w
u u
1 0
= − z b s
−
∂x 3h2
∂x](https://image.slidesharecdn.com/92989-250428105930-5e021169/85/Characterization-Testing-Measurement-and-Metrology-1st-Edition-Chander-Prakash-32-320.jpg)
Characterization, Testing, Measurement, and Metrology 1st Edition Chander Prakash
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- 6. Manufacturing Design and Technology Series Series Editor: J. Paulo Davim This series will publish high- quality references and advanced textbooks in the broad area of manufacturing design and technology, with a special focus on sustainabil- ity in manufacturing. Books in the series should find a balance between academic research and industrial application. This series targets academics and practicing engineers working on topics in materials science, mechanical engineering, indus- trial engineering, systems engineering, and environmental engineering as related to manufacturing systems, as well as professions in manufacturing design. Drills Science and Technology of Advanced Operations Viktor P. Astakhov Technological Challenges and Management Matching Human and Business Needs Edited by Carolina Machado and J. Paulo Davim Advanced Machining Processes Innovative Modeling Techniques Edited by Angelos P. Markopoulos and J. Paulo Davim Management and Technological Challenges in the Digital Age Edited by Pedro Novo Melo and Carolina Machado Machining of Light Alloys Aluminum, Titanium, and Magnesium Edited by Diego Carou and J. Paulo Davim Additive Manufacturing Applications and Innovations Edited by Rupinder Singh and J. Paulo Davim For more information about this series, please visit: https://www.routledge.com/ Manufacturing-Design-and-Technology/book-series/CRCMANDESTEC
- 7. Characterization, Testing, Measurement, and Metrology Edited by Chander Prakash, Sunpreet Singh, and J. Paulo Davim
- 8. First edition published 2021 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2021 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright. com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,978-750-8400.ForworksthatarenotavailableonCCCpleasecontactmpkbookspermissions@ tandf.co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Names: Prakash, Chander, editor. | Singh, Sunpreet, editor. | Davim, J. Paulo, editor. Title: Characterization, testing, measurement, and metrology / edited by Chander Prakash, Sunpreet Singh, J. Paulo Davim. Description: First edition. | Boca Raton : CRC Press, 2020. | Series: Manufacturing design and technology | Includes bibliographical references and index. Identifiers: LCCN 2020018334 (print) | LCCN 2020018335 (ebook) | ISBN 9780367275150 (hardback) | ISBN 9780429298073 (ebook) Subjects: LCSH: Materials—Testing. Classification: LCC TA410 .C457 2020 (print) | LCC TA410 (ebook) | DDC 620.1/10287—dc23 LC record available at https://lccn.loc.gov/2020018334 LC ebook record available at https://lccn.loc.gov/2020018335 ISBN: 978-0-367-27515-0 (hbk) ISBN: 978-0-429-29807-3 (ebk) Typeset in Times by codeMantra
- 9. v Contents Preface vii ...................................................................................................................... Editors ix ....................................................................................................................... Contributors xi .............................................................................................................. Chapter 1 Wear Measuring Devices for Biomaterials . 1 Manoj Mittal ......................................... Chapter 2 Closed Form Solution for the Vibrational Analysis of Metal–Ceramic-Based Porous Functionally Graded Plate . 15 Yogesh Kumar, Ankit Gupta, and Dheer Singh ............... Chapter 3 Functionally Graded Structures: Design and Analysis of Tailored Advanced Composites 33 Ankit Gupta .......................................................... Chapter 4 Heat Transfer in Noncircular Microchannels Using Water and Ethylene Glycol as Base Fluids with Zn and ZnO Nanoparticles 57 Monoj Bardalai, Md. Adam Yamin, Bhaskarjyoti Gogoi, and Rajeev Goswami ................................................................. Chapter 5 Influence of Layer Thickness and Orientation on Tensile Strength and Surface Roughness of FDM-Built Parts. 77 Praveen Kumar Nayak, Anshuman Kumar Sahu, and Siba Sankar Mahapatra ...................... Chapter 6 Effect of Process Parameters on Cutting Forces and Osteonecrosis for Orthopedic Bone Drilling Applications. 93 Atul Babbar, Vivek Jain, Dheeraj Gupta, Chander Prakash, Sunpreet Singh, and Ankit Sharma ............... Chapter 7 Fabrication and Machining Methods of Composites for Aerospace Applications. 109 Atul Babbar, Vivek Jain, Dheeraj Gupta, Chander Prakash, and Ankit Sharma ...................................................................
- 10. vi Contents Chapter 8 Computational Analysis of Aerodynamics Characteristics of High-Speed Moving Vehicle 125 Pawan Singh, Vibhanshu Chhettri, and Nitin Kumar Gupta ............................................................ Chapter 9 Role of Finite Element Analysis in Customized Design of Kid’s Orthotic Product. 139 Harish Kumar Banga, Parveen Kalra, R. M. Belokar, and Rajesh Kumar .................................................... Chapter 10 An Experimental Study of Mechanical and Microstructural Properties of AA7075-T6 by Using UWFSW Process 161 Akash Sharma, V.K. Dwivedi, and Shahabuddin ..................... Index 187 ......................................................................................................................
- 11. vii Preface This book presents a comprehensive treatise of various operational principles, scientific tools, technical methodologies, and qualitative/quantitative characteristics involved in the scientific world. It is comprised four sections: (i) mechanical testing to reveal information about a material’s mechanical properties under dynamic or static forces, (ii) metrological investigations involved to observe the geometrical and structural integrity of the engineering products using optical and analogue calipers, (iii) characterization of the engineering products using spectroscopic tools, and (iv) routine observations that play a vital role in quality assurance. And, measure- ments made within an academic institution, manufacturing facility, or research and development center must be able to be reproduced accurately anywhere in the world. Therefore, the ultimate aim of this book is to understand the reliability aspects of the testing procedures, analyzed products, and quality issues of such procedures in order to understand their industrial implications. In addition, due to the overwhelm- ing growth of the optics-based measurement protocols, characterization techniques such as metallography (light microscopy), X-ray diffraction, transmission and scan- ning electron microscopies, and the theoretical concept strength by using such pro- tocols have been described. We have also incorporated the research activities on the principles of advanced characterization and testing, including the importance of performance-based speci- fications in the manufacturing sector. Apart from the physical examination of the developed engineering products, measurement and testing of same via modeling and simulation also play a critical role to understand the design features and therefore have been incorporated in this edited book.
- 13. ix Editors Chander Prakash is Associate Professor at the School of Mechanical Engineering, Lovely Professional University, Jalandhar, India. He received a Ph.D. in Mechanical Engineering from Panjab University, Chandigarh, India. His areas of research is biomaterials, rapid prototyping and 3D printing, advanced manufacturing, mod- eling, simulation, and optimization. He has more than 11years of teaching expe- rience and 6years of research experience. He has contributed extensively to titanium- and magnesium-based implant literature with publications appearing in Surface and Coating Technology, Materials and Manufacturing Processes, Journal of Materials Engineering and Performance, Journal of Mechanical Science and Technology, Nanoscience and Nanotechnology Letters, and Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture. He has authored 150 research papers and 30 book chapters. He is also an editor of 15 Books: He is also a guest editor of two journals: Special Issue on “Metrology in Materials and Advanced Manufacturing,” Measurement and Control (SCI indexed) and Special Issue on “Nano-Composites and Smart Materials: Design, Processing, Manufacturing and Applications” of Advanced Composites Letters. Sunpreet Singh is researcher in NUS Nanoscience & Nanotechnology Initiative (NUSNNI). He received a Ph.D. in Mechanical Engineering from Guru Nanak Dev Engineering College, Ludhiana, India. His area of research is additive manu- facturing and application of 3D printing for development of new biomaterials for clinical applications. He has contributed extensively to the subject of additive man- ufacturing with publications appearing in Journal of Manufacturing Processes, Composite Part: B, Rapid Prototyping Journal, Journal of Mechanical Science and Technology, Measurement, International Journal of Advance Manufacturing Technology, and Journal of Cleaner Production. He has authored 10 book chapters and monographs. He is working in joint collaboration with Prof. Seeram Ramakrishna, NUS Nanoscience & Nanotechnology Initiative and Prof. Rupinder Singh, Manufacturing Research Lab, GNDEC, Ludhiana. He is also an editor of three books: Current Trends in Bio-manufacturing, Springer Series in Advanced Manufacturing, Springer International Publishing AG, Gewerbestrasse 11, 6330 Cham, Switzerland., December 2018; 3D Printing in Biomedical Engineering, Book series Materials Horizons: From Nature to Nanomaterials, Springer International Publishing AG, Gewerbestrasse 11, 6330 Cham, Switzerland, August 2019; and Biomaterials in Orthopaedics and Bone Regeneration - Design and Synthesis, Book series: Materials Horizons: From Nature to Nanomaterials, Springer International Publishing AG, Gewerbestrasse 11, 6330 Cham, Switzerland, March 2019. He is also Guest Editor of three journals: Special Issue on “Functional Materials and Advanced Manufacturing,” Facta Universitatis, Series: Mechanical Engineering (Scopus Index), Materials Science Forum (Scopus Index), and Special Issue on “Metrology in Materials and Advanced Manufacturing,” Measurement and Control (SCI indexed).
- 14. x Editors J. Paulo Davim received a Ph.D. in Mechanical Engineering in 1997, a M.Sc. degree in Mechanical Engineering (materials and manufacturing processes) in 1991, a Mechanical Engineering degree (five years) in 1986 from the University of Porto (FEUP), the Aggregate title (Full Habilitation) from the University of Coimbra in 2005, and a D.Sc. from London Metropolitan University in 2013. He is Senior Chartered Engineer by the Portuguese Institution of Engineers with an MBA and Specialist title in Engineering and Industrial Management. He is also Eur Ing by FEANI-Brussels and Fellow (FIET) by IET-London. Currently, he is a professor at the Department of Mechanical Engineering of the University of Aveiro, Portugal. He has more than 30years of teaching and research experience in manufacturing, mate- rials, and mechanical and industrial engineering, with special emphasis in machin- ing and tribology. He has also interest in management, engineering education, and higher education sustainability. He has guided many postdoc, Ph.D. and master’s stu- dents as well as coordinated and participated in several financed research projects. He has received several scientific awards. He has worked as evaluator of projects for the European Research Council (ERC) and other international research agencies as well as examiner of Ph.D. candidates for many universities in different countries. He is the editor-in-chief of several international journals, a guest editor of journals, books series, and a scientific advisor for many international journals and confer- ences. Presently, he is an editorial board member of 30 international journals and acts as reviewer for more than 100 prestigious Web of Science journals. He has also published as editor (and co-editor) of more than 100 books and as author (and co-author) of more than 10 books, 80 book chapters, and 400 articles in journals and conferences (more than 250 articles in journals indexed in Web of Science core collection/h-index 49+/7000+ citations, SCOPUS/h-index 56+/10000+ citations, Google Scholar/h-index 70+/16000+).
- 15. xi Contributors Atul Babbar Mechanical Engineering Department Thapar Institute of Engineering and Technology Patiala, India Dr. Harish Kumar Banga Department of Production & Industrial Engineering Punjab Engineering College Chandigarh, India Monoj Bardalai Department of Mechanical Engineering Tezpur University Assam, India Dr. R.M. Belokar Department of Production & Industrial Engineering Punjab Engineering College Chandigarh, India Vibhanshu Chhettri Department of Mechanical Engineering DIT University Dehradun, India V.K. Dwivedi GLA University Mathura, India Bhaskarjyoti Gogoi Department of Mechanical Engineering Tezpur University Assam, India Rajeev Goswami Department of Mechanical Engineering Tezpur University Assam, India Ankit Gupta Shiv Nadar University Dadri, India Dheeraj Gupta Mechanical Engineering Department Thapar Institute of Engineering and Technology Patiala, India Nitin Kumar Gupta Department of Mechanical Engineering DIT University Dehradun, India Vivek Jain Mechanical Engineering Department Thapar Institute of Engineering and Technology Patiala, India Dr. Parveen Kalra Department of Production & Industrial Engineering Punjab Engineering College Chandigarh, India Dr. Rajesh Kumar Department of Mechanical Engineering UIET, Panjab University Chandigarh, India Yogesh Kumar Shiv Nadar University Dadri, India Siba Sankar Mahapatra National Institute of Technology Rourkela Rourkela, India
- 16. xii Contributors Manoj Mittal Department of Mechanical Engineering IKG Punjab Technical University Jalandhar, India Praveen Kumar Nayak National Institute of Technology Rourkela Rourkela, India Chander Prakash School of Mechanical Engineering Lovely Professional University Phagwara, India Anshuman Kumar Sahu National Institute of Technology Rourkela Rourkela, India Shahabuddin GLA University Mathura, India Akash Sharma GLA University Mathura, India Ankit Sharma Chitkara College of engineering Chitkara University Patiala, India Dheer Singh Shiv Nadar University Dadri, India Pawan Singh Department of Mechanical Engineering DIT University Dehradun, India Sunpreet Singh School of Mechanical Engineering Lovely Professional University Phagwara, India Md. Adam Yamin Department of Mechanical Engineering Tezpur University Assam, India
- 17. 1 1 Wear Measuring Devices for Biomaterials Manoj Mittal IKG Punjab Technical University Jalandhar 1.1 WEAR OF BIOMATERIALS: HOW IT IS DIFFERENT FROM OTHER MATERIALS The failure of body implant may be due to wear of one of the joining parts in the body environment. The wear in the implant and surrounding bone may be abrasive, adhesive or fatigue. Some wear testing devices are used to simulate the wear in body environment. Using these wear testing machines, a highly accelerated wear environ- ment (as compared to actual body environment) is created for a shorter time span than actual service life of implant. By doing so, the best suited implant material may be recommended by testing the same in laboratories (in vitro). Wear of biomaterials is measured using specially designed simulators. These simulators simulate actual body environment, and original body implants are tested in these machines for lakhs of cycles in a shorter time span than the actual life span of implant. This is done by highly increasing the number of cycles per unit time. Knee simulator is one of the most complex wear testing devices, which tests actual knee prostheses, and is very costly. Some commercial equipment may cost as expensive as $200,000 or even more. Wang et al. (1999) and Blunn et al. (1991) have reported a comparatively less expensive substitute for knee simulators to exam- ine materials used for knee replacement. Other simpler and inexpensive models for wear testing are pin-on-flat (Van Citters, 2004), pin-on-disc, flat-on-disc and cyclic CONTENTS 1.1 Wear of Biomaterials: How It Is Different from Other Materials 1 .................... 1.2 Machines and Equipment for Biomaterials: Under Dry Conditions and in Simulated Body Environment Joint Simulators 4 ............................................ 1.2.1 Six-Station Rolling Sliding Tribo-Tester 5 .............................................. 1.2.2 How-Medica Biaxial Line Contact Wear Machine 6 .............................. 1.2.3 Three-Axis Knee Wear Simulator 7 ........................................................ 1.2.4 Fein Focus X-Ray Microscope and Fretting Wear Apparatus. 8 ............. 1.2.5 Wear Testing Equipment .9 ...................................................................... 1.3 Conclusions. 11 .................................................................................................... 1.4 Future Scope 11 ................................................................................................... References 11 ................................................................................................................
- 18. 2 Characterization, Testing, Measurement, and Metrology sliding wear testing machines. Morks et al. (2007) utilized SUGA abrasion tester, which follows NUS-ISO-3 standards (Japan) for testing wear on bio-inserts; a similar wear testing machine is built to calculate the wear resistance of coated specimens by Mittal (2012). The release of debris due to wear and subsequent tissue inflammatory response has emerged as a central problem, restraining the long-term clinical outcome of total hip replacements (Harris, 1995; MaGee et al., 1997; Wroblewski, 1997; Raimondi, 2001). The key wear mechanisms observed on retrieved knee prosthesis include delamination caused by surface damage of polyethylene, surface pitting, third body wear and adhesive wear (Blunn et al., 2009; Hood et al., 1983; Landy and Walker, 1988; Collier et al., 1991). Low conformity designs were found to be the cause of delamination (Engh et al., 1992), whereas in a relative conforming design, surface damage was found to be associated with entrapped acrylic particles (Hood et al., 1983). The damage may be because of the different kinematic conditions occurring at bearing surface (Blunn et al., 2009). Excessive sliding led to delamination wear, whereas rolling or cyclic loading at the same contact point resulted in minimal wear (Blunn et al., 1991). Three body wear and production of polyethylene and metallic debris gener- ally occur mainly at articulating joint; however, a little may occur at femoral stem. Mechanical stresses generated by patient on hip implant are supposed to be the foun- dation of the third body wear. There is a mixed response on the effectiveness of hydroxyapatite (HA) coatings in preventing third body wear. Several clinical studies have revealed that HA coatings had no adverse effects; however, other clinical stud- ies have discovered excessive wear at the polyethylene surface due to the accumula- tion of calcium phosphate and metal particles due to third body wear (Sun et al., 2001). Shearing micro-movements may take place at implant and bone interface due to a large difference in elastic modulus of two materials in contact. Insufficient initial fixation (problem in prosthesis design) or movement of limb (which sustains many stresses) in some course of time can also cause micro-movement (Fu, 1999; Walker et al., 1987; Riues et al., 1995). The oscillatory micro-movements at the contact induce fretting wear, fretting corrosion and sometimes fretting cracks, causing early failure of joint prosthesis (Hoppner and Chandrasekaran, 1994; Lambardi et al., 1989). In an investigation by Gross and Babovic (2002), abrasion resistance of coat- ings was ascertained with pin-on-disc arrangement under unlubricated conditions. A bone analogue made of wood, with φ6.3, was used as pin to simulate cortical bone in terms of hardness and elastic modulus. The result of the investigation showed a weight loss of coating and decrease in surface roughness, and main change in the surface characteristics occurred in first minute of testing (Gross and Babovic, 2002). Coathup et al. (2005) inserted six different types of hip replacements (36 in total) into the right hip of skeletally mature female mule sheep and revealed that HA-coated implants were more effective than other uncemented and cemented implants in resisting progressive osteolysis along the acetabular cup–bone interface and also concluded the importance of the in-growth surface present on implant. HA-coated porous acetabular implants showed significant results in terms of bone contact and
- 19. 3 Wear Measuring Devices for Biomaterials in-growth in the presence of wear debris and in the prevention of interfacial wear particle migration. Kalin et al. (2003) studied the wear of the hydroxyapatite pins against glass- infiltrated alumina submerged in a static bath of distilled water at room tempera- ture and reported that wear of hydroxyapatite pin against glass-infiltrated alumina occurred primarily by fracture and deformation. The hydroxyapatite wear particu- lates are mixed with wear products from glass infiltrate in alumina and water to form an intermediate surface layer. Because of this adhered debris layer, steady-state wear is more appropriately described as three-body wear as opposed to two-body wear. Pin-on-disc experiments with HA pins and glass-infiltrated alumina (in-Ceram alumina) conducted by Kalin et al. (2002) showed that the wear volume of HA increased as surface roughness of glass-infiltrated alumina and load was increased, while for a given surface roughness value, the wear factor remained independent of load. Furthermore, polished glass-infiltrated surface showed no evidence for mate- rial transfer at low load, whereas mechanical wear with removal of glass infiltrate was observed at higher loads. Morks et al. (2007) investigated the role of arc current in plasma spray tech- nique on abrasion behavior of coatings and reported that with an increase in arc current, the abrasion resistance of HA coating increases mainly due to the increase in hardness of coating. The resistance to abrasion wear was found to be dependent on coating thickness because the abrasion wear resistance increased as the thickness of HA coating becomes less than 30 μm due to the increase in hard- ness of thin HA coatings. Morks and Kobayashi (2006) studied the dependence of gas flow rate on plasma-sprayed HA coatings and reported that HA coatings sprayed at high flow rates exhibit higher abrasive wear resistance compared to those sprayed at low gas flow rate due to higher cohesion bonding among the splats and low porosity. The wear resistance of HA can be enhanced by reinforcing the secondary phase to HA to produce composite coatings. Several researchers have used various rein- forcement materials such as silica (SiO2), titania (TiO2) (Morks, 2008), alumina (Al2O3), zirconia (ZrO2), carbon nanotubes (CNTs), diamond-like carbon, P2O5– CaO glass, yttria-stabilized zirconia (YSZ) (Balani, 2007), Ni3Al, and titanium and its alloys. A composite powder of HA with 4wt.% multi-walled CNTs was deposited on Ti-6Al-4V. Both HA and HA-CNT composite coatings showed bet- ter wear resistance than Ti-6Al-4V substrate, whereas HA-CNT composite coat- ings result in reduced weight and volume loss in comparison with HA coatings and Ti-6Al-4V substrate. Low weight loss of HA-CNT coating during wear was due to the under-propping and self-lubricating nature of CNTs and the pinning of wear debris assisted by CNT bridging and stretching (Balani et al., 2007). The enhance- ment in wear resistance (Tercero et al., 2009) was observed by reinforcing CNT to HA; furthermore, the resistance to wear was increased by increasing the content of CNT from 0% to 20% and this behavior (Chen, 2007) might be attributed due to the increased hardness, strength, and fracture toughness of composite coatings (Lahiri, 2011) compared to pure HA coating (Chen et al., 2007). Alumina offers a very high wear resistance at articulating surface in orthopedic applications due to
- 20. 4 Characterization, Testing, Measurement, and Metrology its high hardness, low coefficient of friction and excellent resistance to corrosion (Cordingley et al., 2003). Wang et al. (2005) had examined the wear properties with respect to partially stabilized ZrO2 reinforcement to HA against UHMWP in human plasma lubrication and reported improvement in resistance to wear might be due to the addition of reinforced particles. In HVOF-sprayed HA/TiO2 com- posite, mutual reaction could be the cause of chemical bonding between HA and titanium splats. The chemical bonding was found to be beneficial for the prevention of release of titanium particles as wear debris which can lead to prosthesis rejection or infection (Li et al., 2002). 1.2 MACHINES AND EQUIPMENT FOR BIOMATERIALS: UNDER DRY CONDITIONS AND IN SIMULATED BODY ENVIRONMENT JOINT SIMULATORS Joint simulators are standard testing equipment used to measure tribological behav- ior of actual joint used in repair and replacements of broken and damaged bone and tissues in an artificially created environment. They mimic the orientation and magnitude of the load of actual joint. Ideal simulators are sealed and temperature controlled, and produce a similar type of wear with a similar rate of wear debris that is of comparable morphology as found in actual clinical cases. After completing millions of cycles in a simulator (which take a considerable time), specimens are weighed and the difference in weight loss is measured by comparing with the initial weight of the specimen. A simulator, in general, does not provide quantitative data directly. The wear is also inspected with a scanning electron microscope. These simulators only account for physical tribological factors and do not take into con- sideration the biological factors that would affect the tribology in a body. Biological factors may be taken care of by introducing simulating/artificial body environment in the simulator. The choice of lubricant to be used in joint simulator is another important fac- tor to be considered. Joint simulator requires a relatively large quantity of lubricant fluid. The human joint fluid is not available due to its limited supply, as healthy knees do not contain much fluid and due to inflammation, the knees have enough synovial fluid for suitable extraction. The research studies thus relegated to use syn- thetic lubricants; American Society for Testing and Materials (ASTM) simply rec- ommends that these lubricants be volume, concentration and temperature controlled (Hirakawa et al., 2004). ASTM recommends a bovine serum lubricant supplemented with 0.2%–0.3% sodium azide and 20 ml ethylenediaminetetraacetic acid (EDTA). The sodium azide limits bacterial growth in the fluid and has a role in inhibiting protein adsorption on the surface of implant, whereas EDTA discourages calcium phosphate precipitation. Overall, a joint simulator provides a reasonably good idea regarding wear performance for artificial joints. The time taken for analysis, vari- ability of lubricants, and, foremost, the cost limit their use. Unfortunately, the cost of commercial models of knee simulator exceeds $200,000 (₹10 million). The most complex wear testing devices are knee simulators, which test actual knee prosthesis. Most of the knee simulators provide physiologic loading and can reproduce at least four out of six degrees of freedom: flexion/extension (F/E),
- 21. 5 Wear Measuring Devices for Biomaterials anterior/posterior (A/P), tibial rotation and abduction/adduction. Some of the joint simulators are discussed in detail hereunder. 1.2.1 Six-Station Rolling Sliding tRibo-teSteR The six-station rolling sliding tribo-tester is a quite effective machine (Kennedy et al., 2006) for testing the contact fatigue behavior of different orthopedic poly- mers (Van Citters et al., 2004) and is equally useful in testing their wear behavior due to clinically relevant stress and motion environments. The tribo-tester articu- lates cobalt–chromium cylinders against UHMWPE pucks (Kennedy, 2006). The Hertzian line contact theory is utilized to determine contact stresses resulting from specimen geometry. Moreover, the non-confirming contact creates surface subsur- face stresses that are similar to those found in contemporary total knee replacement bearings (Kennedy et al., 2000 and Van Citters et al., 2004). Alignment of the articu- lating cylinders is performed by placing pressure-sensitive films at contact between a Co-Cr cylinder and an UHMWPE puck. The macrograph of rolling sliding tribo- tester is represented in Figure 1.1. FIGURE 1.1 Multi-station rolling sliding tribo-tester: (a) macrograph of tribo-tester, (b) con- tact conditions in one of the stations (Kennedy et al., 2007) and (c) CAD model of one of the six stations. UHMWPE specimen is attached to upper shaft, whereas metallic specimen is with lower shaft. The contact region is contained in a sealed fluid chamber (Van Citters, 2004).
- 22. 6 Characterization, Testing, Measurement, and Metrology The intensity of color on otherwise translucent film reflects contact pressure at that location. Shims are generally inserted in the bearing mounts to make the Co-Cr cylinder and UHMWPE puck parallel, and by doing so, uniform contact pressure profile is created. After the insertion of shims, all six stations should have the same distribution of pressure across the contact width. Each station has a fluid bath, which provides lubrication and cooling at contact points. The contact can be viewed through the acrylic windows incorporated with the system. The design of the testing equip- ment ensures that all contact axes are normal to direction of sliding. Each pair can be loaded independently as per the requirement of test as constant load pneumatic system is incorporated with each pair of cylinder and puck. A crank and rocker arm powered by a fixed-speed motor is used for articulation. Backlash is prevented by adjustable gears as well as adjustable chains and sprockets. The degree of sliding can be adjusted by combinations of sprockets in the drive train. The two test specimens can oscillate and rotate with different amplitudes and speeds. The relative oscillation can be altered for each pair of specimens independently as per the requirement of test. Linear wear can be determined by using profilometer. The measurements should be taken at the earliest after the test has been completed to avoid any viscoelastic relaxation. A six-station knee simulator is represented in Figure 1.2. 1.2.2 How-Medica biaxial line contact weaR MacHine A wear machine with two parallel axes for evaluation of wear of implant bearing materials for total knee joint replacement was developed by Wang et al. (1999). The design used by the researchers is based on simulating both geometry and motion of knee joint. FIGURE 1.2 Three stations of AMTI-Boston six-station knee simulator, Model KS2–6– 1000 (AMTI, 200).
- 23. 7 Wear Measuring Devices for Biomaterials A static load of 1150N is applied for simplicity as well as to reduce cost. A rotating Co-Crring(ϕ71.9××25mm)isloadedagainstUHMWPEblock(1.6cm××1.6cm××1.25cm), whichoscillatesaboutloadaxistosimulatetibialrotation.Thedegreeofflexion-extension is 0°–60°, and that of tibial rotation is 0°–30°. Bovine serum is used to lubricate and cool the contact points of the specimens. The limitation of this machine is the absence of anterior-posterior sliding and physiological correct loading. The static loading may decrease the wear rate. The wear testing equipment described here, which is very low in cost as compared to commercial six-station knee simulator manufactured by MTC (Eden Prairie, MN, USA) and Instron-Stanmore (Canton, MA, USA), shows parallel results of wear rates and wear mechanism in terms of morphology of wear surface and wear products. The macrograph of two of the twelve stations of How-Medica biaxial line contact wear machine (Wang et al., 1999) is represented in Figure 1.3. 1.2.3 tHRee-axiS Knee weaR SiMulatoR The three-axis knee simulator with ball-on-surface contact was designed and built by Saikko et al. (2001) for basic wear and friction tests of materials used in knee prosthesis. The motion of machine consists of flexion-extension (F/E), anterior- posterior (A/P) translation and inward-outward (I/O) rotation. The F/E is applied by ball and the A/P and I/O with the disc, as represented in Figure 1.4a. A crank mechanism is utilized to implement motion, and variation with time is nearly sinusoidal with a cycle time of 0.93s. A large Co-Cr ball represents femo- ral component, and a flat polyethylene disc, located horizontally beneath the ball, represents tibial component, as represented in Figure 1.4b. The frame, motor and loading system of an old five-station, uniaxial hip joint simulator was utilized to complete one station of the three-axis knee wear simulator (and Saikko et al., 2001). FIGURE 1.3 Two stations of twelve-station How-Medica biaxial line contact wear machine (Wang et al. 1998).
- 24. 8 Characterization, Testing, Measurement, and Metrology The 54mm diameter Co-Cr ball with 42.4° F/E motion was used to revolve on the disc, and a vertical upward static load of 2 kN was applied on the lower side of the disc, as shown in Figure 1.4. The detailed design of machine is reported by Saikko et al. (2001). 1.2.4 Fein FocuS x-Ray MicRoScope and FRetting weaR appaRatuS The equipment consists of a Fein Focus X-ray image processing system (Fein Focus Roentgen-System, Germany) with a beam spot size 4–20μm in diameter (Fu et al., 1998). Micro-focused X-rays directly penetrate contact surfaces and form an evident image on the fluorescent screen placed below the specimen. A high-speed digital camera captures the images developed on fluorescent screen and converted them to visual images, which can be displayed on a monitor. A defi- nite time interval is selected to capture the images, and image processing functions are used to treat and enhance the images to reveal the detailed phenomenon of fret- ting wear. The line diagram of the equipment is represented in Figure 1.5. This equipment is an improved version of simple ball-on-flat fretting wear testing machine. The upper specimen is a ball made from hardened stainless steel AISI 410 with a diameter of 25mm, as shown in Figure 1.5. The material to be investigated for wear studies is made in form of a flat and mounted on a table which reciprocates at a specified frequency. Electromagnetic FIGURE 1.4 Three-axis knee simulator with ball-on-flat contact: (a) principle and (b) close-up.
- 25. 9 Wear Measuring Devices for Biomaterials exciters are used to oscillate the table at a given frequency. A function generator controls the displacement via a power amplifier (Fu, 1999). A piezoelectric force transducer is used to measure tangential frictional force, and the amplitude of oscil- latory movement is measured with a laser sensor with a minimum resolution of 2μm (Fu et al., 1998). 1.2.5 weaR teSting equipMent All the joint simulators which simulate motions and loading of the actual human knee joint are very expensive for procurement and very difficult to handle and man- age. Many investigators have used some simple machines to estimate the wear loss of the coated metallic as well as bare metallic specimens. Khan et al. (1996), Balani et al. (2007), Kalin et al. (2003) and Kalin et al. (2002) investigated wear resis- tance of specimens on pin-on-disc wear testing equipment using phosphate-buffered saline, simulated body fluid, carboxymethyl cellulose in distilled water, and dis- tilled water, which is used to lubricate the contact, respectively, whereas Gross and Babovic (2002), Sidhu et al. (2007) and Bolelli et al. (2006) conducted wear experi- ments under dry conditions on pin-on-disc wear tester. Lima et al. (2006), Morks and Kobayashi (2006), Morks et al. (2007) and Morks and Akimoto (2008) used flat-on-disc without any lubricant to conduct wear experiments of coated specimens. The pin-on-disc apparatus allows a pin to articulate on a smooth disc, which spins in a rotary manner beneath the pin including relative sliding motion between pin and disc. Normal load is applied, and frictional force in the transverse direction and tangent to circular track is obtained. This device provides both wear and frictional data. The pin can be weighted for loss of mass over a set period of revolutions or distance slid. The contact of pin and disc can be lubricated and cooled using some FIGURE 1.5 Schematic illustration of fretting wear tribometer (Fu et al., 1998).
- 26. 10 Characterization, Testing, Measurement, and Metrology lubricant as discussed earlier. The reverse combination of pin and disc can also be used in which pin is made of harder material such as zirconia and disc is the speci- men for which the wear resistance is to be measured. In this arrangement, disc can be weighed for any loss of mass. A schematic illustration of pin-on-disc apparatus is represented in Figure 1.6. Karanjai et al. (2008) studied the fretting wear on Ti-Ca-P bio-composite against bearing steel over 10,000 cycles at a frequency of 10Hz and stroke of 80μm with 2–10N load under dry conditions and in Simulated Body Fluid (SBF), and reported that under dry conditions, the steady-state coefficient of friction increases with load, whereas in SBF, it is independent of load and have a very low value when compared to dry conditions. The fretting wear rates were found to be less by two orders of mag- nitudes in SBF than those under dry conditions at low loads. Under dry conditions, the wear rate decreased at high loads due to the formation of a tribo-layer, whereas it increased with load in SBF medium. The flat-on-disc allows a flat surface to slide on the periphery of a disc, which revolves beneath the flat, as represented in Figure 1.7. The rpm of the disc can be changed by means of gear attachment. A servo motor is incorporated to slide the test sample as well as to rotate the disc about its axis. A vertical load is applied. The weight loss of flat specimen is measured to calculate the wear if the disc is an abra- sive wheel, whereas the weight loss of flat and disc is measured if both are coated. As shown in Figure 1.7 in SUGA abrasion testing equipment, the 1cm wide wheel is covered with 400 grit SiC emery paper, which moves at 25 mm/min. The flat slides 1cm on the disc with a normal load of 10N. The flat specimen and the disc have a line contact of 1cm (Morks et al., 2007, 2008; Morks and Akimoto, 2008). The SUGA wear tester follows NUS-ISO-3 standard (Japan). Inspired by the design which offers better wear measurement due to line con- tact, combination of sliding and rotational motion, which reproduces two degrees FIGURE 1.6 Schematic illustration of pin-on-disc wear machine with lubrication facility (Balani et al., 2007).
- 27. 11 Wear Measuring Devices for Biomaterials of motion, i.e., anterior-posterior and flexion-extension (flat sliding specimen and rotating disc) out of six degrees of motion of knee joint, similar equipment was built. Flannery et al. (2008) simulated the same two degrees of motion in three-station wear simulators to measure wear of tibial inserts. 1.3 CONCLUSIONS Various types of wear testing machines are discussed in detail, and it was found that for actual body implants, high-end multi-axes simulators with many degrees of freedoms are very expensive and can only be utilized for testing of real-time body implants in manufacturing industries. Other similar low-cost machines are suitable for laboratory work. 1.4 FUTURE SCOPE Different types of wear testing equipment and machines are extensively used by researchers as well as in medical industry. In these machines and equipment, small- scale replica of actual implant with the same or similar materials is tested in an aggressive environment. Real implants need to be tested, and machines are being developed which can measure life span of real implants in an artificial real-like envi- ronment and different loading conditions of various limbs of human body. New and better reinforcements are added to increase the service life of implant. REFERENCES AMTI.biz. Balani, K., Anderson, R., Laha, T., Andara, M., Tercero, J., Crumpler, E. and Agarwal, A., (2007), “Plasma-sprayed carbon nanotube reinforced hydroxyapatite coatings and their interaction with human osteoblasts in vitro”, Biomaterials, Vol. 28, pp. 618–624. Balani, K., Chen, Y., Harimkar, S. P., Dahotre, N. B. and Agarwal, A., (2007) “Tribological behavior of plasma sprayed carbon nanotube-reinforced hydroxyapatite coating in physiological solution”, Acta Biomater., Vol. 3, pp. 944–951. FIGURE 1.7 Schematic illustration of flat-on-disc wear tester (SUGA wear tester) (Morks et al., 2007).
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- 31. 15 2 Closed Form Solution for the Vibrational Analysis of Metal– Ceramic-Based Porous Functionally Graded Plate Yogesh Kumar, Ankit Gupta, and Dheer Singh Shiv Nadar University 2.1 INTRODUCTION Functionally graded (FG) materials are the superior tailored hybrid materials in which the properties of the material vary gradually in the preferred direction. Extensive literature has been published in the area of FG structures. In this context, Zhang et al. [1] presented an extensive review based on the buckling and vibrational response of the functionally graded plate (FGP). Gupta et al. [2–4] analyzed the vibrational behavior of a porous gradient plate using hybrid higher-order shear deformation theory (HSDT). Kanuet et al. [5] reviewed on smart functionally graded materials (FGMs) and analyzed buckling and vibration analysis based on fracture problems. Kurtaran et al. [6] studied the vibrational response of FGP and showed the effects of vibrating shape. Thai et al. [7] explored the vibrational and bending analysis of FGP using simple HSDT. Gupta and Talha [8] have studied and analyzed the effect of porosity on the structural response of a FGP using HSDT. They have reported that the fundamental frequency of the porous plate increases as the volume fraction CONTENTS 2.1 Introduction 15 .................................................................................................... 2.1.1 Mathematical Formulation 16 .................................................................. 2.1.1.1 Displacement Field 16 ............................................................... 2.1.1.2 Energy Equations 18 ................................................................. 2.1.1.3 Constitutive Relation 21 ............................................................ 2.1.1.4 Analytical Solution 23 .............................................................. 2.2 Results and Discussion 25 ................................................................................... 2.3 Conclusions 31 ..................................................................................................... References. 31 ...............................................................................................................
- 32. 16 Characterization, Testing, Measurement, and Metrology index (VFI) increases. Significant changes have been reported in frequency for thin and thick plates. Shahverdi and Barati [9] have analyzed the free vibrational response of porous FGP. To investigate the vibrational response of a porous FGP on an elastic substrate, a nonlocal strain gradient elastic model has been developed. The authors have reported that the influence of various parameters such as nano-pores, nonlocal parameters, and gradient index have influence on fundamental frequencies of FGP. Kumar et al. [10] proposed and carried out the geometrically non-linear analysis of the FGP using HSDT. Zenkour et al. [11] used the 3D elasticity solution for an Exponentially functionally graded plate (E-FGP). Birman V. et al. [12] proposed the research article on the modeling and investigation of FG structures. Reddy et al. [13] emphasized the comprehensive investigation of FGP. Benveniste et al. [14] suggested a new methodology for the use of Mori-Tanaka’s principle FGP. Mori T et al. [15] reported average stresses in the average elastic energy in various materials. Shariat et al. [16] presented the stability analysis of FGP with geometrical imper- fection based on Classical plate theory (CPT). While using FSDT, Lanhe [17] derived the equilibrium and stability equations for a simply supported rectangular FGP with moderate thickness under thermal loads. Javaheri at al. [18,19] obtained the equilibrium and stability equation using variational approach method which is based on the CPT for perfect simply supported FGP. The authors have also derived the equilibrium equations under thermal loads based on the HSDT and power-law gradient rule. M Mohammadi et al. [20] developed a method to decouple the stability equation of moderately thick FGP using levy-type boundary conditions. An efficient and simply refined plate theory for studying buckling behavior of FGP using CPT was developed by Thai et al. [21]. Shariat et al. [22,23] introduced the rectangular FGP with or without geometric defects for mechanical and thermal analyses with the assumption of non-homogeneous mechanical properties varying through the plate thickness under the compression, tension, and thermal loading. In this research work, the algebraic-based HSDT has been considered for the vibrational analysis of FGP. The power-law distribution and Hamilton’s variational principle have been used to derive the governing equations. Governing equations are solved using Navier’s method. The effects of various parameters like VFI and geometric parameter on the fundamental frequency of FGP have been reported in subsequent section. The model of the porous FGP has been shown in Figure 2.1. The material proper- ties of the porous FGP are shown in Table 2.1. 2.1.1 MatHeMatical FoRMulation 2.1.1.1 Displacement Field The displacement field, i.e., u i i for 1 = ,2,3, is the function of x, y, z, and t used in this work for the functional grade plate, reported by H.T. Thai [7], and expressed as the following equations: ∂w 4z3 ∂w u u 1 0 = − z b s − ∂x 3h2 ∂x
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