The free vibration behavior of a bidirectional functionally graded rotating micro-disk that is subjected to uniform transverse pressure and high-temperature thermal loading has been studied. The micro-disk is functionally graded along the radial and thickness directions. The problem is mathematically formulated using two distinct but interrelated steps within the framework of Kirchhoff plate theory and modified couple stress theory. The first step determines the time-invariant deformed configuration of the micro-disk under centrifugal, pressure, and thermal loading using minimum potential energy principle. The second step determines the free vibration behavior of the micro-disk in the neighborhood of the deformed configuration using Hamilton’s principle. The solutions of the governing equations for both these steps are obtained using the Ritz method. The mathematical model is successfully validated with various reduced problems. The numerical results for the first four axisymmetric bending vibration modes are presented to investigate the effects of wide range of parameters such as rotational speed, applied pressure, thermal loading, size-dependent thickness, volume fraction indices, and radius ratio. The mode shapes of vibration are illustrated through surface and contour plots.

The present work studies the free vibration response of functionally graded rotating micro-disks subjected to transverse pressure and thermal loading based on the modified couple stress theory. The disk material is considered to be functionally graded along the radial and thickness directions, and its properties are assumed to be temperature-dependent following the Touloukian model. The mathematical formulation is based on an energy functional involving the von Kármán type non-linearity, in which appropriate displacement derivatives and its conjugate stress measures are used to define the strain energy of the micro-disk. The minimum potential energy principle is employed to develop the governing equations for determining the deformed configuration of the micro-disk under combined centrifugal, pressure and thermal loading. Further, the governing equations for free vibratory motion of the micro-disk are derived following Hamilton’s principle and incorporating the tangent stiffness of the deformed micro-disk. The governing equations are discretized and solved employing the Ritz method. The mathematical model is successfully validated with different reduced problems available in the literature. The influence of rotational speed, transverse pressure, thermal loading, size-dependent thickness and volume fraction indices are investigated for a wide range of parametric values. Some illustrative mode shapes along with the contour have also been presented. The present study is first of its kind and the presented results would definitely serve as benchmarks for any further study in this field.

The free vibration behavior of a statically deflected Timoshenko microbeam under uniformly distributed static load is studied based on modified couple stress theory. In the first step of the analysis, the beam configuration under large static deflection is obtained through a nonlinear static analysis in which the governing equations are derived employing minimum potential energy principle and incorporating von Karman geometric nonlinearity. In the subsequent step, the free vibration behavior of the statically deflected microbeam is investigated employing Hamilton’s principle and incorporating the tangent stiffness of the statically deflected beam configuration. The solutions of the governing equations for both the steps are obtained by approximating the displacement fields following Ritz method. The model is validated using the available results in the literature for some reduced problems. The results for the first two vibration modes are presented in nondimensional frequency–amplitude plane for clamped, simply supported, and clamped–simply supported beams.

A mathematical model for investigating the asymmetric as well as the axisymmetric free vibration behavior of a rotating annular micro-disk is presented for the first time. The disk is assumed to be functionally-graded (FG) along the radial and thickness directions, and is considered to be operating in high-temperature environment. An energy based approach involving minimum potential energy principle and Hamilton's principle is used to derive the governing equations of motion considering Kirchhoff plate theory. The size-effect is addressed employing modified couple stress theory. A novel tangent stiffness based formulation is employed to derive the governing equations of vibratory motion in the neighborhood of the centrifugally and thermally deformed disk configuration. The governing equations are solved following Ritz method. The model captures both the axisymmetric and asymmetric flexural vibration modes, as well as the torsional mode. The model is successfully validated with the available results for some reduced problems. Numerical results are presented in tabular and graphical form for various material and geometric parameters, and some illustrative mode shape plots are presented showing the mode-switching phenomenon. The work presents a generalized model which can be reduced to theoretically model a wide variety of practical problems.

Nano-Ag/DLC/cellulose composite free-standing films were deposited by using hybrid chemical routes. Cellulose fibers were extracted from banana pseudostem fibers by chemical route. Modulation of bonding environment on cellulose upon nano-Ag and diamond-like carbon (DLC) inclusions was studied by X-ray photoelectron spectroscopy (XPS), Raman and Fourier transformed infrared spectroscopy (FTIR) measurements. Changes in microstructural and bonding environmental properties of the cellulose films with nano-Ag and DLC loadings were also addressed. These composite films would harness the development of anti-bacterial and bio-compatibility in futuristic bandage applications. Antibacterial studies using Staphylococcus aureus (gram positive) and Escherichia coli (gram negative) bacteria were carried out on the above n-Ag/DLC/cellulose composite films.

An improved mathematical model of pre-twisted tapered rotating micro beams made of bidirectional functionally graded material (BFGM) is presented to study its free vibration behavior. The effects of spin-softening and Coriolis acceleration are incorporated, and modified couple stress theory is employed to address the size effect. The mathematical formulation is based on first-order shear deformation theory and is developed in a global non-inertial frame incorporating appropriate transformations between the global inertial frame and the local non-inertial frame. Two different but interrelated steps are employed, where the first step determines the centrifugally deformed configuration using minimum potential energy principle, and the second step determines the free vibration behavior through tangent stiffness of the deformed rotating beam using Hamilton's principle. The direct use of tangent stiffness considers the centrifugal stiffening effect through von Kármán non-linearity and bypasses the need of strain energy functional for the vibrating beam. The governing equations are transformed to an eigenvalue problem through state-space approach and solved following Ritz method. The effects of spin-softening, Coriolis acceleration, and pre-twist angle are shown and discussed. The effects of different parameters such as size-dependent parameter, aspect ratio, material gradation indices, operating temperature, FGM constituent, taperness parameters, and slender parameter are discussed.

Free vibration behavior of rotating bidirectional functionally graded nano-beams is studied based on Eringen’s nonlocal theory. The beam material consists of ceramic and metal constituents, and the material is graded across the length and thickness directions. The mathematical formulation is framed on Euler–Bernoulli beam theory, and the system of governing equations is derived in variational form using Hamilton’s principle. The governing equations are discretized and transformed to an eigen value problem using Ritz method. The model is formulated to study the flapping and lead-lag motions due to free vibration. The model is verified with the available numerical results. The numerical results are presented in non-dimensional frequency-speed plane to study the influence of normalized nonlocal parameter, length gradient parameter, thickness gradient parameter, root radius parameter, and section aspect ratio. Some normalized mode shapes are presented to illustrate the mode switching phenomenon. The mathematical model of a nonlocal rotating bidirectional functionally graded material nano-beam is presented for the first time through this work and the reported results are new of its kind.

A mathematical model based on Eringen's nonlocal elasticity theory is presented to analyze free vibration behavior of rotating nano-beams. The model is capable of studying the flap-wise and chord-wise vibrations as well as the axial vibration of rotating nano-beams. The model is based on Euler-Bernoulli beam theory, and incorporates spin-softening and Coriolis effects. Hamilton's principle is employed to derived the governing equations involving virtual displacements, and Ritz method is followed to discretize the governing equations. The governing equations are transformed to an eigenvalue problem in state-space. The stable solutions in frequency domain are indentified by appropriately examining the nature of the complex eigenvalues. The model is validated through some reduced problems available in the literature. The non-dimensional speed-frequency behaviors are presented and discussed for different normalized nonlocal parameters, hub parameters and section aspect ratios. The spin-softening and Coriolis effects are individually illustrated and discussed. The present advanced model for rotating nano-beams is reported for the first time. The present study would help in understanding the dynamics of rotating nano-beams in a comprehensive manner.

An improved mathematical model is presented to investigate the free vibration behavior of post-buckled tapered functionally graded material beam, subjected to uniform temperature rise and steady-state heat conduction. The material properties including the thermal conductivity are considered to be temperature-dependent and an iterative algorithm for solving temperature-dependent steady-state heat conduction equation is presented to get the correct temperature profile. The initial static post-buckling problem is formulated using minimum potential energy principle and the subsequent free vibration problem is formulated using Hamilton’s principle by employing the tangent stiffness of the post-buckled configuration. The solution of the governing equations is obtained using Ritz method. Following Timoshenko beam theory, a geometrically non-linear mathematical model is developed by employing the non-linear strain–displacement relationships for both normal and shear strains. The study is carried out for both hinged–hinged and clamped–clamped beams. Non-dimensional load–frequency behaviors are presented for different gradation indices, taperness parameters, and length–thickness ratios. Static post-buckling equilibrium path for clamped–clamped beams is also presented. The significant effects of shear non-linearity and temperature-dependent thermal conductivity on dynamics of tapered functionally graded material beam are shown in the paper.

Geometrically nonlinear forced vibration analysis of higher order shear-deformable functionally graded microbeam is presented, where the beam is supported on a three-parameter Winkler–Pasternak-type nonlinear elastic foundation and subjected to a harmonically varying distributed load. The modified couple stress theory of elasticity is employed in the formulation to address the size-dependent effect. Hamilton’s principle is used to derive the displacement-based governing equations considering Reddy’s third-order shear deformation theory. Ritz method is followed to convert the governing equations to nonlinear algebraic form in the frequency domain by approximating the displacement fields. A mixed algorithm for nonlinear equations based on the iterative substitution method with successive relaxation and Broyden’s method is successfully employed to solve the stable regions of the frequency-response curves. The results are presented for hinged and clamped beams, and the effects of different parameters such as size-dependent thickness, load amplitude, foundation parameters, and gradation-profile parameter are studied. The effect of thermal loading due to uniform temperature rise is also studied considering temperature-dependent material properties.

Free vibration behavior of bidirectional-functionally graded, double-tapered rotating micro-beam is investigated. An improved mathematical model based on Timoshenko beam theory and modified couple stress theory is developed that includes the effects of geometric non-linearity, spin-softening, Coriolis acceleration and high operating temperature. The problem is formulated in two steps. In the first step, the problem involving time-invariant inertia force due to rotation of the beam with constant angular speed is formulated using minimum potential energy principle and the governing non-linear equations are solved employing an iterative algorithm. In the next step, the free vibration problem is formulated employing Hamilton’s principle and using the tangent stiffness of the deformed configuration induced due to time-invariant inertial loading. The governing equations for free vibration are transformed to state-space to formulate an eigenvalue problem. The governing equations are solved by approximating the displacement fields following Ritz method. The model is successfully validated with the available results. Extensive sets of results are presented for the first two chord-wise and flap-wise modes of vibration in non-dimensional speed versus frequency plane. The effects of different parameters such as size-dependent thickness, axial and thickness gradation indices, taperness parameters, hub parameter, length-thickness ratio, operating temperature and FGM composition are reported.

Geometrically non-linear forced vibration analysis of higher-order shear-deformable functionally graded material beam under harmonic excitation and supported on three-parameter non-linear elastic foundation is presented. The beam is immovably clamped and is considered to be under static thermal loading due to uniform temperature rise. Reddy’s third-order shear-deformable beam theory in conjunction with von Kármán geometric non-linearity is considered to derive the governing equations employing Hamilton’s principle, and Ritz method is followed for approximating the displacement and rotation fields. A numerical algorithm based on iterative substitution method and Broyden’s method is proposed to predict the stable regions of frequency-response behavior. The frequency-response curves are presented in normalized plane for variations of load-amplitude, elastic foundation parameters, temperature rise, gradation index and functionally graded material composition, and their effects are discussed in detail. It is found that the load-amplitude, elastic foundation parameters, thermal loading and some of the functionally graded material compositions significantly affect the frequency response; whereas, the effect of gradation index is found to be relatively small. A comparative frequency-response curve between Voigt model and Mori–Tanaka scheme of functionally graded material modeling is presented, and it shows negligible difference between these two models. The present problem under thermal environment is studied for the first time through this work, and the proposed model and the numerical algorithm provide a simplified approach to study the non-linear frequency-response behavior.

For thermally postbuckled configurations, the free vibration behavior of functionally graded (FG) Timoshenko beams are investigated. The postbuckling configurations are obtained through a geometrically nonlinear static problem. The free vibration problem around the postbuckled configuration is formulated using its tangent stiffness. The energy based governing equations are solved following the Ritz method. The elements of the tangent stiffness matrix are obtained using the Ritz coefficients. The results are shown to exhibit the effects of FG material, material profile parameter, and length-thickness ratio. The comparative results are presented for both the cases of the physical neutral surface and the geometrical neutral surface.

The present work investigates the free vibration behavior of double-tapered functionally graded beams rotating in thermal environment, using an improved mathematical model. The functional gradation for ceramic–metal compositions, following power-law, is considered to be symmetric with respect to the mid-plane, leading to metal-rich core and ceramic-rich outer surfaces of the beam. The temperature dependence of the material properties are considered using Touloukian model. The nonlinearity in strain–displacement relationships for both the axial and transverse shear strains are considered. Firstly, the governing equations for deformed beam configuration under time-independent centrifugal loading are obtained using minimum total potential energy principle, and the solution is obtained following Ritz method. Then the free vibration problem of the centrifugally deformed beam is formulated employing Lagrange’s principle and considering tangent stiffness of the deformed beam configuration. Coriolis effect is considered in the mathematical model, and the governing equations are transformed to the state-space for obtaining an eigenvalue problem. The results for the first two modes of both chord-wise and flap-wise vibrations are presented in nondimensional plane to show the effects of taperness parameter, root-offset parameter, volume fraction exponent, operating temperature, and functionally graded material composition. The results in comparative form are presented for both temperature-dependent and temperature-independent material properties.

Cellulose fibers were obtained from banana pseudo-stems and subjected to different treatments to achieve nanocrystals. The cellulose materials obtained after different stages of extraction were studied using X-ray diffraction (XRD), X-ray photoelectron spectroscopy (XPS), Raman spectroscopy and Fourier transform infrared spectroscopy (FTIR). XRD measurements revealed higher crystalline cellulose content in the acid hydrolyzed sample than in the raw fiber, with crystallite sizes of similar to 3.5 nm. XPS results indicated a significant intensity decrease of the peak located at similar to 285 eV, corresponding to the non-cellulosic component, in the spectrum of the raw fiber, as compared to that of the acid hydrolyzed sample Raman studies indicated higher content of sp(2) hybridized carbon in the raw fiber, due to its high amorphous carbon content, while acid treated crystalline samples showed significantly higher sp(3) hybridized carbon content, indicating higher crystallinity. FTIR studies indicated that the raw material contained the highest amount of sp(2) bonded carbon and the acid treated crystalline samples contained the most sp(3) bonded carbon.

An improved mathematical model to study the free vibration behavior of rotating functionally graded material beam is presented, considering non-linearity up to second order for the normal and transverse shear strains. The study is carried out considering thermal loading due to uniform temperature rise and using temperature-dependent material properties. Power law variation is assumed for through-thickness symmetric functional gradation of ceramic–metal functionally graded beam. The effects of shear deformation and rotary inertia are considered in the frame-work of Timoshenko beam theory. First, the rotating beam configuration under time-invariant centrifugal loading and thermal loading is obtained through a geometrically non-linear analysis, employing minimum total potential energy principle. Then, the free vibration analysis of the deformed beam is performed using the tangent stiffness of the deformed beam configuration, and employing Hamilton’s principle. The Coriolis effect is considered in the free vibration problem, and the governing equations are transformed to the state-space to obtain the eigenvalue problem. The solution of the governing equations is obtained following Ritz method. The validation is performed with the available results, and also with finite element software ANSYS. The analysis is carried out for clamped-free beam and for clamped–clamped beam with immovably clamped ends. The results for the first two modes of chord-wise and flap-wise vibration in non-dimensional speed-frequency plane are presented for different functionally graded material compositions, material profile parameters, root offset parameters and operating temperatures.

A comprehensive study on different physico-chemical properties of banana fiber is presented in this communication. This has led to a better understanding of its structure and function. Studies were carried out for both inner and outer region of inflorescence stem. FESEM studies clearly indicated that the inner-wall of the micro tubules consisted of closely spaced helical xylem fibers forming a cylindrical shape attached to it. XRD studies indicated that both the samples contained crystalline cellulose and high amount of amorphous content. The XPS peak located at ∼533 eV could be attributed to cellulose and hemicelluloses. The FTIR bands at ∼2925 cm −1 corresponded C-H stretching vibration in cellulose component. Fibers from outer and inner regions showed nearly similar cellulose content.

The present work investigates the out-of-plane (or flap-wise) and in-plane (or chord-wise) free bending vibration of FGM beam, which is fixed to the inside of a rotating rim. A general formulation considering non-linearity for both normal and shear strains is presented employing Timoshenko beam theory. A symmetric through-thickness material gradation following power law distribution of volume fraction is considered. The first step of the problem determines the beam configuration under time-invariant centrifugal loading through a geometrically non-linear analysis, employing minimum total potential energy principle. The second step of the problem determines the free vibration frequency of the beam about the deformed configuration, and in that case, the tangent stiffness of the beam configuration is used to formulate the governing equations employing Hamilton’s principle. The effects of spin-softening and Coriolis acceleration components are considered. The solution of the governing equations is obtained following Ritz method, and the free vibration problem is transformed to a standard eigenvalue problem by transforming the equations to the state-space. The critical speed leading to buckling of the rotating beam is reported. The results are presented in non-dimensional speed-frequency plane for different root offset parameters, volume fraction indices and FGM compositions.

The present work aims at the determination of thermal buckling loads of various functionally graded material beams with both ends clamped. Thermal loading is applied by applying linear temperature distribution and nonlinear temperature distribution at steady state heat conduction condition, across the beam thickness. Temperature dependences of the material properties, considered in the formulation, make the present problem physically nonlinear. Also, the effect of limit thermal load at which the effective elastic modulus and/or thermal expansion coefficient become theoretically zero is considered. The mathematical formulation is based on Euler–Bernoulli beam theory. An energy based variational principle is employed to derive the governing equations as an eigenvalue problem. The solution of the governing equation is obtained using an iterative method. The validation of the present work is carried out with the available results in the literature and with the results generated by finite element software ANSYS. Four different functionally materials are considered, namely, stainless steel/silicon nitride, stainless steel/alumina, stainless steel/zirconia, and titanium alloy/zirconia. Comparative results are presented to show the effects of variations of volume fraction index, length–thickness ratio, and material constituents on nondimensional thermal buckling loads.