Multimode Nonlinear Vibration Analysis of Stiffened Functionally Graded Double Curved Shells in a Thermal Environment

The motivation of the current work is to develop a multi-modal analysis of the nonlinear response of stiffened double curved shells made of functionally graded materials under thermal loads. The formulation is based on the first order shear deformation shell theory in conjunction with the von Kármán geometrical nonlinear strain-displacement relationships. The nonlinear equations of motion of stiffened double curved shell based on the extended Sanders’s theory were derived using Galerkin’s method. The resulting system of infinite nonlinear ordinary differential equations, that includes both cubic and quadratic nonlinear terms, was solved using a nonlinear dynamic software XPPAUT to obtain the force-amplitude relationship. The effect of both, longitudinal and transverse stiffeners, was considered using the Lekhnitsky’s technique and the material properties are temperature dependent and vary in the thickness direction according to the linear rule of mixture. In order to obtain accurate natural frequency in thermal environments, critical buckling temperature differences are carried out, resulting in closed form solutions. The effect of temperature’s variation as well as power index, functionally graded stiffeners, geometrical parameters, temperature depended materials and initial imperfection on the nonlinear response of the stiffened shell are considered and discussed. This dissertation showed that the nonlinear study of problems of thin-walled structures with even stiffeners is of paramount importance. It was also found that the difference between single-mode and multi-mode analyses could be very significant for nonlinear problems in a thermal environment. Hence, multimode vibration analysis is necessary for structures of this nature

Thermal buckling of functionally graded piezomagnetic micro- and nanobeams presenting the flexomagnetic effect

Galerkin weighted residual method (GWRM) is applied and implemented to address the axial stability and bifurcation point of a functionally graded piezomagnetic structure containing flexomagneticity in a thermal environment. The continuum specimen involves an exponential mass distributed in a heterogeneous media with a constant square cross section. The physical neutral plane is investigated to postulate functionally graded material (FGM) close to reality. Mathematical formulations concern the Timoshenko shear deformation theory. Small scale and atomic interactions are shaped as maintained by the nonlocal strain gradient elasticity approach. Since there is no bifurcation point for FGMs, whenever both boundary conditions are rotational and the neutral surface does not match the mid-plane, the clamp configuration is examined only. The fourth-order ordinary differential stability equations will be converted into the sets of algebraic ones utilizing the GWRM whose accuracy was proved before. After that, by simply solving the achieved polynomial constitutive relation, the parametric study can be started due to various predominant and overriding factors. It was found that the flexomagneticity is further visible if the ferric nanobeam is constructed by FGM technology. In addition to this, shear deformations are also efficacious to make the FM detectable

Mechanics of Micro- and Nano-Size Materials and Structures

For this reprint, we intend to cover theoretical as well as experimental works performed on small scale to predict the material properties and characteristics of any advanced and metamaterials. New studies on mechanics of small-scale structures such as MEMS/NEMS, carbon and non-carbon nanotubes (e.g., CNTs, Carbon nitride, and Boron nitride nanotubes), micro/nano-sensors, nanocomposites, macrocomposites reinforced by micro-/nano-fillers (e.g., graphene platelets), etc., are included in this reprint

Non-linear forced vibration study of axially functionally graded non-uniform beams by using Broyden method

In the forced vibration case due to external loading deflection tends to higher value so beam element undergoes the non-linearity behavior. Basically non linearity is categorized into two parts material and geometric non-linearity. Due to large amplitude we have to take care of both in-plane and out-plane displacement fields. With the help of variation form of energy principle all displacement fields calculated. These displacement fields are the combination of admissible orthogonal function. Admissible functions satisfies the both flexural and Membrane boundary conditions. By considering non-linear strain displacement relationship geometric nonlinearity is introduced in this thesis. The resulting nonlinear set of governing equations is solved through a numerical procedure involving direct substitution method using relaxation parameter. So calculation of natural frequency under both free and forced vibration decides the working condition of beam. Now a day in structural field tremendous growth is taking place for which we required a material which attains a good property and it became possible by functionally graded materials. Functionally graded material is non-homogenous and anisotropic material whose both structural and material property varies along the element. In FGM effect of residual stress and stress concentration is minimum in between two dissimilar materials which increase the strength and toughness of that structural element. So in this paper we are dealing with forced vibration analysis of FGM. For the static analysis we are using minimum potential energy principle and for the dynamic analysis we are using Hamilton’s principle. Further non-linearity of beam can be calculated by using Broyden method. Through this research we obtained some results which are validated with some previous papers and then we submitted our further research results

Large deformation behavior of functionally graded porous curved beams in thermal environment

The in-plane thermoelastic response of curved beams made of porous materials with different types of functionally graded (FG) porosity is studied in this research contribution. Nonlinear governing equations are derived based on the first-order shear deformation theory along with the nonlinear Green strains. The nonlinear governing equations are solved by the aid of the Rayleigh–Ritz method along with the Newton–Raphson method. The modified rule-of-mixture is employed to derive the material properties of imperfect FG porous curved beams. Comprehensive parametric studies are conducted to explore the effects of volume fraction and various dispersion patterns of porosities, temperature field, and arch geometry as well as boundary conditions on the nonlinear equilibrium path and stability behavior of the FG porous curved beams. Results reveal that dispersion and volume fraction of porosities have a significant effect on the thermal stability path, maximum stress, and bending moment at the crown of the curved beams. Moreover, the influence of porosity dispersion and structural geometry on the central radial and in-plane displacement of the curved beams is evaluated. Results show that various boundary conditions make a considerable difference in the central radial displacements of the curved beams with the same porosity dispersion. Due to the absence of similar results in the specialized literature, this paper is likely to provide pertinent results that are instrumental toward a reliable design of FG porous curved beams in thermal environment

THERMAL BUCKLING AND BENDING ANALYSES OF CARBON FOAM BEAMS SANDWICHED BY COMPOSITE FACES UNDER AXIAL COMPRESSION

The bending and critical buckling loads of a sandwich beam structure subjected to thermal load and axial compression were simulated and temperature distribution across sandwich layers was investigated by finite element analysis and validated analytically. The sandwich structure was consisted of two face sheets and a core, carbon fiber and carbon foam were used as face sheet and core respectively for more efficient stiffness results. The analysis was repeated with different materials to reduce thermal strain and heat flux of sandwich beams. Applying both ends fixed as temperature boundary conditions, temperature induced stresses were observed, steady-state thermal analysis was performed, and conduction through sandwich layers along with their deformation nature were investigated based on the material properties of the combination of face sheets and core. The best material combination was found for the reduction of heat flux and thermal strain, and addition of aerogel material significantly reduced thermal stresses without adding weight to the sandwich structure

Natural vibration induced parametric excitation in delaminated Kirchhoff plates

This paper revisits the problem of free vibration of delaminated composite plates with Lévy type boundary conditions. The governing equations are derived for laminated Kirchhoff plates including through-width delamination. The plate is divided into two subplates in the plane of the delamination. The kinematic continuity of the undelaminated part is established by using the system of exact kinematic conditions. The free vibration analysis of orthotropic simply supported Lévy plates reveals that the delaminated parts are subjected to periodic normal and in-plane shear forces. This effect induces parametric excitation leading to the susceptibility of the plates to dynamic delamination buckling during the vibration. An important aspect is that depending on the vibration mode the internal forces have a two-dimensional distribution in the plane of the delamination. To solve the dynamic stability problem the finite element matrices of the delaminated parts are developed. The distribution of the internal forces in the direction of the delamination front was considered. The mode shapes including a half-wave along the width of the plate accompanied by delamination buckling are shown based on the subsequent superimposition of the buckling eigenshapes. The analysis reveals that the vibration phenomenon is amplitude dependent. Also, the phase plane portraits are created for some chosen cases showing some special trajectories

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”

Mechanical Instability of Thin Solid Film Structures

Instability of thin film structures as buckling and wrinkling are important issues in various fields such as skin aging, mechanics of scars, metrology of the material properties of thin layers, coating of the surfaces and etc. Similar to the buckling, highly ordered patterns of wrinkles may be developed on the film‒substrate due to compressive stresses. They may cause a failure of the system as structural damage or inappropriate operation, however once they are well understood, it is possible to control and even use them properly in various systems such as the gossamer structures in the space, stretchable electronics, eyelike digital cameras and wound healing in surgery. In this thesis, the mechanical instability of the thin film is considered analytically and numerically by solving the eigenvalue problem for the governing equation of the system, and the effects of the different factors on the instability parameters such as load, amplitude, wavenumber and length of the wrinkles are studied. Different problems such as wrinkling within an area on the film, and buckling and wrinkling of the non‒uniform systems with variable geometry and material properties for both of the film and substrate are investigated. It is shown that the effects of the non‒uniformity of the system are very significant in localization of the wrinkles on the film; however, such a factor has been ignored by many researchers to simplify the problems. In fact, for the non‒uniform systems, the wrinkles accumulate around the weakest locations of the system with lower stiffness and the wrinkling parameters are highly affected by the non‒uniformity effects. Such effects are important especially in thin film technology where the thickness of the film is in the order of Micro/Nano scale and the uniformity of the system is unreliable. The results of this dissertation are useful in the design and applications of thin films in science, technology and industry. They consider the relation of the loading and structural stiffness with the wrinkling parameters and provide more insight into the physics of the localization of the wrinkling on the thin structures, how and why wrinkles are accumulated at some positions. Therefore, deliberate application of these results provides appropriate tools to control and use the buckling and wrinkling of thin films effectively in different fields

Effect of axial porosities on flexomagnetic response of in-plane compressed piezomagnetic nanobeams

We investigated the stability of an axially loaded Euler–Bernoulli porous nanobeam considering the flexomagnetic material properties. The flexomagneticity relates to the magnetization with strain gradients. Here we assume both piezomagnetic and flexomagnetic phenomena are coupled simultaneously with elastic relations in an inverse magnetization. Similar to flexoelectricity, the flexomagneticity is a size-dependent property. Therefore, its effect is more pronounced at small scales. We merge the stability equation with a nonlocal model of the strain gradient elasticity. The Navier sinusoidal transverse deflection is employed to attain the critical buckling load. Furthermore, different types of axial symmetric and asymmetric porosity distributions are studied. It was revealed that regardless of the high magnetic field, one can realize the flexomagnetic effect at a small scale. We demonstrate as well that for the larger thicknesses a difference between responses of piezomagnetic and piezo-flexomagnetic nanobeams would not be significant