Front Matter

The Editor thanks all the contributors for their efforts and support in preparing and submitting their chapters on time. I greatly appreciate all the researchers and authors mentioned in the references sections of this book who have directly or indirectly helped us to develop this book in a systematic way.

The Editor would like to thank his parents, the late Sri B. K. Chakraborty and the late Sri Mati Parul Chakraborty, for their blessings. Furthermore, he would like to thank and appreciate his wife, Shewli, and daughters, Shreyati and Susprihaa, for their love, support, and being sources of inspiration throughout.

The authorities at NIT Rourkela are gratefully acknowledged for their moral and additional support. The fruitful comments and suggestions of the reviewers are also sincerely acknowledged.

Finally, the Editor wishes to thank the entire AIP Publishing team for their help and support throughout this project toward the success of the book and its timely publication.

Snehashish Chakraverty

Editor

The application of nanomaterials has recently been increasing among the science and technology communities of physics, chemistry, engineering, nanotechnology, biology, sensors, computation, and so forth. These materials possess special mechanical, electrical, and electronic properties. As a result, nanomaterials have been shown to play significant roles in various nanoelectromechanical systems and nanocomposites. These nanomaterials may be in the form of nanoparticles, nanowires, nanotubes, nanotube resonators, nano actuators, and so forth, and such structures are becoming very important in civil, mechanical, and aerospace engineering. Therefore, the static and dynamic analyses of nanostructures are of great interest to various researchers. Appropriate knowledge about mechanical behavior is essential for a better design of these materials. However, conducting experiments on the nanoscale is both complicated and expensive. Consequently, the development of appropriate mathematical models and their solutions to study the static and dynamic behavior of nanostructures is significant and important. These models/problems are governed by linear/nonlinear differential equations, which are not always able to be solved analytically. This deficiency then compels us to search for various numerical/-computational methods. In this regard, this book is an honest attempt to compile the computationally efficient methods and a few experimental ideas into one place to understand the static and dynamic behaviors of nanostructures.

This book contains eleven chapters written by different authors. Chapter 1 introduces the basics of nanostructures, and the static problems are presented in Chaps. 2–4. Viscosity measurement for picoliter liquid volumes is addressed in Chap.5 and a molecular dynamics problem is targeted in Chap. 6. Finally, Chaps. 7–11 contain dynamic problems.

Chapter 1 has been written by Karan Kumar Pradhan and Snehashish Chakraverty, where an introduction to nanostructures is provided. This chapter discusses the origin and gradual development of nanostructures in various disciplines of science and technology. Furthermore, brief discussions on the nanostructural members have been provided.

Multiscale modeling for the static effects of nanostructures is addressed in Chap. 2 contributed by Kim-Quang Hoang and Chennakesava Kadapa. To investigate the mechanical properties of novel materials, the recently proposed method of multiscale virtual power (MMVP) is adapted to derive finite element formulations for the micro-discrete to macro-continuum transition based on a discrete representative volume element (RVE) at the nanoscale. The interactions among the atoms are modeled using nonlinear structural finite elements. The potential of the proposed multiscale framework for the characterization of nanomaterials has been demonstrated using numerical examples consisting of single-layer graphene, with and without defects.

In Chap. 3, Ömer Civalek, Hayri Metin Numanoğu, Shahriar Dastjerdi, and Bekir Akgöz explore the static bending analysis of small-sized structures using various non-classical theories. Here, the size-dependent static bending response of microbeams is investigated based on the modified couple stress and modified strain gradient elasticity theories. The equilibrium equations with classical and non-classical boundary conditions for microbeams are derived by implementing the principle of minimum total potential energy based on the modified couple stress and modified strain gradient theories in conjunction with Bernoulli–Euler beam theory. The resulting higher-order equation is analytically solved for various boundary conditions.

Static bending of an Euler nanobeam with surface effects is considered in Chap. 4, which is presented by Somnath Karmakar and Snehashish Chakraverty. The governing equation is derived with the help of Eringen's nonlocal theory. The differential quadrature method is applied to solve the differential equations, and different classical boundary conditions and surface effects are considered to address the titled problem.

In Chap. 5, M. A. Changizi and I. Stiharu have targeted the “viscosity measurement for picoliter liquid volumes”. Specific physical quantities that characterize the fluids are virtually impossible to measure when the fluids are micrometer-sized droplets but in the femtogram weight range. Devices capable of working with such sizes are rare and the influences from the environmental conditions at that level become extremely significant. The scope of this chapter is to set up the principle and the fundamental models that provide the best information about the viscosity of a small volume of fluid. Further, a discussion of the suitable models to measure the viscosity of picoliter range volumes of fluids using the vibration response of a fluid picoliter droplet is provided.

Pokula Narendra Babu, K. Vijay Reddy, and Snehanshu Pal then examine the problem of dynamic structural evolution of nanocrystalline aluminum during ratcheting deformation in Chap. 6. The structural evolution and dislocation nature are critically studied at the atomic level during the ratcheting deformation. The ratcheting deformation mechanism and dislocation behavior at the grain boundary (GB) of nanocrystalline (NC) aluminum (Al) having a grain size of ∼8 nm are investigated by molecular dynamics simulations at various temperatures. The structural evolution, dislocation types, and dislocation–dislocation interactions during the deformation of NC Al have also been studied under uniaxial tensile and ratcheting simulation tests.

Theoretical concepts of nanostructural dynamic problems are detailed in Chap. 7 presented by Karan Kumar Pradhan and Snehashish Chakraverty. The dynamic problems of the nanostructural members viz. nanotubes, nanobeams, nanoplates, nanocomposites, nanoshells, and so forth are governed by different partial differential equations. As such, this chapter provides exhaustive theoretical concepts with respect to the differential equations of various nanostructural members.

In Chap. 8, vibration analysis of a nanostructural member has been investigated by using the Hermite– Ritz method. This chapter is contributed by Subrat Kumar Jena and Snehashish Chakraverty. Here,

the Hermite–Ritz method has been used to investigate the vibration characteristics of the nanobeam exposed to a longitudinal magnetic field and linear hygroscopic environment. The nanobeam is modeled with the Winkler–Pasternak elastic foundation and nonlocal Euler–Bernoulli beam theory. The governing equation of motion of the proposed model has been derived using Hamilton's principle, and non-dimensional frequency parameters for different boundary conditions have been computed.

Laxmi Behera and Snehashish Chakraverty present the problem of vibration and buckling of nanobeams embedded in an elastic medium under the influence of temperature in Chap. 9. The formulation of the problem is based on the Reddy beam theory in conjunction with nonlocal elasticity theory and the problem is solved by using the differential quadrature method. The frequency and critical buckling load parameters are shown to be dependent on the temperature, elastic medium, small-scale coefficient, and length-to-diameter ratio.

Miniature structures (i.e., micro and nanostructures) have received considerable attention in recent years owing to their small size, low weight, simple fabrication, and high-frequency operations. Sometimes, these structures encounter large amplitude vibrations and dynamic chaos during operational conditions. These undesired phenomena often cause unpleasant motion, disturbing noise, and an unacceptable level of dynamic stresses in nanostructural problems which may lead to potential fatigue failure, degraded performance, energy losses, and decreased reliability. Such a detrimental effect needs to be mitigated for the safety, reliability, and continuous operation of a nanodevice or system. Hence, a suitable vibration control strategy can only address this problem by integrating self-controlling and self-monitoring capabilities into the structure. Accordingly, Puneet Kumar, J. Srinivas, and Michael Ryvkin explore vibration control in nanostructural problems in Chap. 10. This chapter is devoted to summarize the various available vibration control techniques for nanostructural problems. A comprehensive insight on active, hybrid, and semi-active vibration control strategies for nanostructures is presented.

Finally, Chap. 11 includes the problem of dynamical characteristics of a nanostructural member with a complicating effect. This chapter has been written by Subrat Kumar Jena and Snehashish Chakraverty. Here, Navier's technique is used to investigate the vibration and buckling characteristics of three different types of single-walled carbon nanotubes (SWCNTs): armchair, chiral, and zigzag, while using a novel non-local elasticity theory and the Euler–Bernoulli beam theory. Governing equations for vibration and buckling are derived by incorporating Hamilton's principle, and non-local stress resultants are considered. A thorough investigation for the influence of several scaling parameters, such as small-scale parameters, temperature, thermal environment, and CNT length, on the fundamental natural frequencies and critical buckling loads has been carried out.

In view of the different challenges mentioned above, it is hoped that the book will be helpful to undergraduates, graduates, researchers, industry, faculties, and so forth, throughout the globe. The book may also be used for teaching classical and advanced nanostructural static as well as dynamics concepts to graduate and undergraduate students, and to researchers. Further, the Editor believes that the contents

of this book contributed by various expert authors will be useful for classical dynamics, nanostructural static/dynamic, and related computational methods courses which are part of B. Tech., M. Tech., and Ph.D. programs of various universities/institutes throughout the world.

The editor thanks all the contributors for their effort and support in preparing and submitting the chapters on time. Last but not the least, the editor also thanks the entire AIP team for their help and support toward the success of this project.

Snehashish Chakraverty

Editor August 2021

Bekir Akgöz

Division of Mechanics, Civil Engineering Department, Akdeniz University, Antalya, Turkey

Pokula Narendra Babu

Metallurgical and Materials Engineering Department, National Institute of Technology Rourkela, Rourkela 769008, Odisha, India

Laxmi Behera

Department of Mathematics, Model Degree College Nabarangpur, Odisha, India

Snehashish Chakraverty

Department of Mathematics, National Institute of Technology Rourkela, Rourkela 769008, Odisha, India

M. A. Changizi

Principal Mechanical Engineer, Plug Power, Latham, NY, 12110, USA

Ömer Civalek

China Medical University, Taichung, Taiwan

Shahriar Dastjerdi

Division of Mechanics, Civil Engineering Department, Akdeniz University, Antalya, Turkey

Kim-Quang Hoang

Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Bay Campus, Swansea SA1 8EN, United Kingdom

Subrat Kumar Jena

Department of Mathematics, National Institute of Technology Rourkela, Rourkela 769008, Odisha, India

Chennakesava Kadapa

School of Engineering, University of Bolton, Bolton BL3 5AB, United Kingdom

Somnath Karmakar

National Institute of Technology Rourkela, Rourkela 769008, Odisha, India

Puneet Kumar

Department of Aerospace Engineering, Karunya Institute of Technology and Science, Coimbatore, Tamil Nadu, India

Hayri Metin Numanoğlu

Civil Engineering Department, Division of Mechanics, Giresun University, Giresun, Turkey

Snehanshu Pal

Metallurgical and Materials Engineering Department, National Institute of Technology Rourkela, Rourkela 769008, Odisha, India

Karan Kumar Pradhan

Department of Basic Science, Parala Maharaja Engineering College, Berhampur, Sitalapalli 761003, Odisha, India

K. Vijay Reddy

Metallurgical and Materials Engineering Department, National Institute of Technology Rourkela, Rourkela 769008, Odisha, India

Michael Ryvkin

The Iby and Aladar Fleischman Faculty of Engineering, School of Mechanical Engineering, Tel Aviv University, Ramat Aviv 69978, Israel

J. Srinivas

Department of Mechanical Engineering, National Institute of Technology Rourkela, Rourkela 769008, Odisha, India

I. Stiharu

Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal, QC, H3G1M8, Canada